論文 / 著書情報
Article / Book Information
題目(和文)
Title(English)

Integrable dynamics on intelacing arrays for $U<sub>q</sub>(\widehat{\mathfrak{sl<sub>2</sub>}})$ stochastic vertex models

著者(和文)

ムッチコニ マッテオ

Author(English)

Matteo Mucciconi

出典(和文)

学位:博士(理学),学位授与機関:東京工業大学,
報告番号:甲第11569号,
授与年月日:2020年9月25日,
学位の種別:課程博士,
審査員:笹本 智弘,齋藤 晋,村上 修一,古賀 昌久,利根川 吉廣

Citation(English)

Degree:Doctor (Science),
Conferring organization: Tokyo Institute of Technology,
Report number:甲第11569号,
Conferred date:2020/9/25,
Degree Type:Course doctor,
Examiner:,,,,

学位種別(和文)

博士論文

Type(English)

Doctoral Thesis

Powered by T2R2 (Tokyo Institute Research Repository)

Doctoral thesis

Integrable dynamics on intelacing arraysc2) stochastic vertex models
for Uq (slMatteo Mucciconi

東京工業大学Tokyo Institute of Technology

Abstract
In the context of randomly fluctuating interfaces in (1+1)-dimensions we consider the
Kardar-Parisi-Zhang universality class. In the last 20 years integrable techniques weredeveloped to study representative models in this class and mathematical predictions havebeen experimentally verified with extreme precision.
The most general integrable model available to study KPZ phenomena is the Higher
Spin Vertex Model (HSVM) and in this thesis we extend the theory of its integrability.
This is accomplished through the study of two families of special symmetric functions: thespin Hall-Littlewood (F) and the spin q-Whittaker (F) functions. Yang-Baxter relations
endow F and F of symmetries and in parallel offer them probabilistic meaning via bijectivization techniques. Random fields weighted by F and F possess Markov projections
giving rise to the HSVM.
Novel eigenrelations for these special functions allow a systematic study of one point
statistics of the fields. From these we recover the 1 : 2 : 3 scaling and characteristic
fluctuations for the HSVM and its various specializations.

Contents
1 Introduction and results
6
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Spin Hall-Littlewood and spin q-Whittaker functions
2.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Spin Hall-Littlewood functions . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Spin q-Whittaker polynomials . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Analytic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Scaled geometric specializations . . . . . . . . . . . . . . . . . . . . . . . .
2.6 Difference operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.7 Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29
29
31
35
44
50
51
62

3 Integrable random fields of Young diagrams
3.1 Integrable stochastic dynamics on interlacing arrays . . . . . . . . . . . . .
3.2 Schur case: Robinson–Schensted–Knuth from Yang–Baxter . . . . . . . . .
3.3 Marginals of spin Hall-Littlewood and spin q-Whittaker fields . . . . . . .
3.4 Random boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Fredholm determinants from q-moment formulas . . . . . . . . . . . . . . .

64
64
73
75
87
90

4 KPZ fluctuation of the Higher Spin Vertex Model97
4.1 Fredholm determinants from elliptic determinants . . . . . . . . . . . . . . 97
4.2 Asymptotic analysis in the stationary regime . . . . . . . . . . . . . . . . . 106
5 Spin Whittaker functions from q → 1 limit
136
5.1 Spin Whittaker functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.2 Spin Whittaker Processes and beta polymers . . . . . . . . . . . . . . . . . 152
5.3 Deformed quantum Toda system . . . . . . . . . . . . . . . . . . . . . . . . 157
A Special functions
163
A.1 q-deformed quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.2 Classical special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

1

B Yang-Baxter equations
167b
B.1 Uq (sl2 ) R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
B.2 Corner R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
B.3 Nonnegativity of terms in the Yang-Baxter equations . . . . . . . . . . . . 176
C Extras180
C.1 Bounds for φl , ψl , Φx , Ψx . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.2 Construction of contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.3 Triangular Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

2

Acknowledgements
I arrived in Japan in September 2016 and I have been in Tokyo Institute of Technology
first as a Research Student and then as a PhD Student. In these four years I haveconducted my research under the supervision of Tomohiro Sasamoto and he is the first
person I would like to thank. He introduced me to the study of the wonderful theory ofIntegrable Probability, he thought me a method to understand and attack problems andhe has constantly encouranged and fostered my ambition to produce valuable and original
research. I will give my best to live up to these standards.
Soon after I arrived in Tokyo, Tomohiro Sasamoto introduced me to his collaborator
and friend Takashi Imamura. Since then Imamura Sensei has been for me a second mentor
and together we have been through countless many difficult calculations and puzzling
problems, the solutions of which will find their way to the press in the coming years. I
would like to thank him for his invaluable help and patience.
In January 2018, at a conference in Melbourne I met Leonid Petrov and since then we
have been collaborators. My dearest thanks go to him too, for giving me the chance towork with him and for spending his time to teach me valuable lessons.
In Melbourne I also met Alexander Garbali and we have since then been friends. I
would like to thank him for being an invaluable resource when in need of clarifications
and discussions about special symmetric functions.
I would also like to express my gratitude towards all members of Sasamoto Sensei’slaboratory. Especially I would like to thank Masato Usui, Rykuo Nagao and Hiroki Moriya
for all the friendly conversations we have had and for helping me improve my japaneseskills.
In December 2018 I became married to Natasha Ikhsan and my final thanks go to her.
She is the strongest and smartest person I know and my highest source of inspiration andI feel priviledged to have the honor to spend the rest of my life with her. Thank to her
for always chosing to be on my side.

3

About this thesis
In 2017 Sasamoto Sensei had showed me a recent discovery by him and Imamura Sensei,
and that was an elegant computation producing exact formulas for a solvable model inthe KPZ class. The model in case was the q-TASEP, a deformation of the classical
Totally Asymmetric Simple Exclusion Process (TASEP). Two years earlier Corwin andPetrov had introduced a rich generalization of the q-TASEP, the higher spin vertex model,that as of today remains the most general exactly solvable model in the KPZ class. A
natural question was whether it was possible to bring techniques invented by Imamura
and Sasamoto at the level of the stochatic vertex models and this thesis contains results
stemming from this investigation.

4

Publication List
[IMS20] Stationary Stochastic Higher Spin Six Vertex Model and q-Whittaker measure. T.Imamura,
M. Mucciconi and T. Sasamoto. Probab. Theory Relat. Fields (2020).
[BMP19] Yang-Baxter random fields and stochastic vertex models. A.Bufetov, M.Mucciconi
and L.Petrov. Advances in Mathematics (to appear).
[MP20] Spin q-Whittaker polynomials and deformed quantum Toda. M.Mucciconi and L.Petrov.
arxiv:2003.14260 (submitted)

5Chapter 1
Introduction and results
1.1

Background

1.1.1

Brief history of KPZ universality

Investigations around universality aspects of growing surfaces have been conducted by
the statistical physics community since at least the early ’80s. It was then known, thanks
to numerical observations and renormalization group arguments, that certain models of
random surfaces allowing lateral growth and possessing smoothening mechanisms exhibit
non trivial relaxations patterns, resulting in the emergence of scaling exponents for correlations and fluctuations. The most eminent representative of such growth models became
the KPZ stochastic partial differential equation, studied first by Kardar, Parisi and Zhang
in 1986 [KPZ86]. For one spatial dimension it reads√
λ∂t H = ν∂x2 H + (∂x H )2 + Dη,
2

(1.1)

where H is the height profile, η is the space-time Gaussian white noise and ν > 0, D > 0, λare non-zero constants1 . In this particular case scaling exponents can be explicitly determined and predictions on the universal phenomena made more precise [KPZ86; HH85]:
this is the case of the so called KPZ universality class, borrowing the name from the modelthat started the field. This class of systems includes directed polymers, random stirred
fluids and many more processes along with growth phenomena, but in general each ofthem can be described by its own height function h(x, t) depending on the time t and on
one spatial coordinate x. In the large space-time limit the process h, under the 1 : 2 : 3characteristic scaling,h(−2 x, −3 t) → 0,(1.2)
is expected to converge, after model dependent deterministic shifts, to a universal fluctuation field, today commonly known as KPZ fixed point [MQR17]. This the strong KPZ
universality conjecture. In fig. 1.1 we compare the KPZ scaling theory with that of weakly
1

A rescaling of height, space and time transforms (1.1) into the same equation with ν = λ = D = 1.

6

KPZ-type evolution

Evolution with no interactions

O(t1/3 )O(t1/2 )

O(t)

O(t)

Figure 1.1: Comparison between two types of random evolution of a surface, both obtained
as large scales of discrete systems. In the left panel the microscopic dynamics follows acorner growth model, defined in section 1.1.2. Fluctuations have size t1/3 and since theinitial profile is flat they obey GOE Tracy-Widom distribution when t  0. Spatial
correlations scale as t2/3 . In the right panel each point of the interface updates f (x) →f (x) + 1 with rate 1. In this case one point fluctuations have size t1/2 and they follow thegaussian distribution.
interacting systems (obeying the gaussian law) to give a taste of the non trivial implications of such predictions; for example one point fluctuations scale as t1/3 and they are notin general governed by the gaussian distribution.
The pivotal realization that KPZ universal fluctuations admit an exact description
came in 1999 from the work of Baik, Deift and Johansson on the Ulam problem [BDJ99].
This asks to characterize the statistics of the longest increasing subsequence in a random
permutation, but it can rephrased as a problem of growth of random surfaces. The
absolutely surprising finding is that the limiting law of the longest increasing subsequenceis given by none other than the Tracy-Widom distribution F2 , that describes fluctuationsof the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) [TW94]. Tracy-Widom
distributions were already known to be universal object in random matrices and more in
general they were regarded as the analog of the gaussian bell curve for strongly interacting
systems.
Stemming from these intriguing developments, the literature of the last twenty years
has presented a constellation of spectacular progresses in characterizing KPZ phenomena.
Further connections with random matrix models came from results of Baik and Rains[BR01] and Sasamoto [Sas05] that related asymptotic law of the Gaussian Orthogonal
Ensemble (GOE) with that of a surface growing off a substrate, while Johansson [Joh00]
studied first the so called droplet initial condition case and related the one point fluctuation with GUE statistics. Growth processes corresponding to other special classes of
initial conditions were also considered. As an example the limiting process produced by
the stationary evolution of a surface was characterized again by Baik and Rains [BR00],
although this was not found to have connections with matrix models. More recently a fulldescription of the KPZ fixed point for general classes of initial conditions became available thanks to Matetski, Quastel and Remenik [MQR17]. We will give a more thoroughaccount of this list of progresses and what made them possible in the following sections,7but we still like to suggest the interested reader to consult [Cor12], [QS15] and references
therein for a detailed exposition on the topic.
The existence of exact methods to determine exponents and fluctuations for a generalclass of growth phenomena carries significant physical implications. The derivation oflaws such as Tracy-Widom statistics is only possible while studying very special exactly
solvable system, therefore experimental proofs of theoretical predictions are of crucial importance. Early attempts to detect universal scaling exponents were done in the late ’90s
studying growing interfaces of mutant bacteria colony [Wak+97] or slow combustion of
paper [Myl+01]. In 2010 breakthrough results of Takeuchi and Sano gave the first precise experimental confirmation of KPZ universal phenomena [TS10], when they observedthat the growing interface of topological-defect turbulence in liquid crystals possesses
characteristic 1:2:3 scaling and GUE Tracy-Widom distribution. By properly triggering
the liquid crystal with a laser beam they were also able to reproduce interfaces growing
off a flat substrate [TS12] and off a brownian motion shaped profile [IFT20], confirming
the emergence respectively of GOE and Baik-Rains statistics and therefore the geometry
dependence of the universal fluctuation field. Additional experimental studies include
cancer colony growth [Hue+12; Hue+10] and the deposition of colloid particles [Yun+11;
Yun+13]; see [Tak14] for a review.
Despite the tremendous successes achieved in the difficult task of pinning down theuniversal scale invariant process describing out of equilibrium one dimensional fluctuatinginterfaces, our understanding of properties and boundaries of KPZ phenomena is far fromsatisfactory. Mathematically exact results such that of Baik-Deift-Johansson are only
available for a handful of models and no technique at the moment has proven successful
in rigorously extending universal laws to more general settings. A full comprehension of
the range of systems encompassed by the theory of KPZ universality is also currently outof reach as literature keeps presenting surprises day after day; see [LŽP19] for a recent
example and [Spo20] for a review.

1.1.2

Stochastic integrable systems and special functions

The major takeaway from the course of events leading to the discovery of Tracy-Widom
laws in growth processes is that a full description of KPZ universal phenomena is accessible thanks to inexplicably nice and rich algebraic structures underlying some of its
representative models. These are the so called stochastic integrable models and theirsystematic study has become known in literature as integrable probability.
The scope of this subsection is to provide the reader a glance at some tools andphilosophy aspects driving the study of integrable probability, especially in the context of
KPZ class. We will do this presenting simple examples with historical relevance, such as
the corner growth model and the Schur processes and doing so we will stress the interplay
between stochastic systems and special symmetric functions coming from representation
theory and algebraic combinatorics.

8

j

g1,5

g2,5

g3,5

g4,5

g5,5

g1,4

g2,4

g3,4

g4,4

g5,4

g1,3

g2,3

g3,3

g4,3

g5,3

g1,2

g2,2

g3,2

g4,2

g5,2

g1,1

g2,1

g3,1

g4,1

g5,1
i

Figure 1.2: A realization of the corner growth model. Red cells lay at corners of theinterface delimiting the gray area and they will be absorbed (become gray) after a waitingtime given by the value of the respective gi,j .
Corner growth model and Schur measure
Among the earliest and simplest models of random growth in one dimension we find thecorner growth model [Ros81]. Decorate the cells of the lattice Z≥0 ×Z≥0 with independent
non-negative random variables {gi,j : i, j ≥ 1}, like in fig. 1.2. In the figure the set ofgray cells represents the growth process, while the thick black down-right path π is theinterface. White cells become active (red) once they are at a corner of the profile π andget absorbed after a waiting time given by the value of the respective gi,j . A simple
quantity we can consider is the first absorption time T (i, j) of a cell (i, j) and this also
turns out to have a simple characterization. Since the cell (i, j) can only be absorbed
if both (i − 1, j) and (i, j − 1) are already in the gray area, the function T satisfies therecursion T (i, j) = gi,j + max{T (i − 1, j), T (i, j − 1)} and more in general we have



 X
T (i, j) = max
gi0 ,j 0 ,(1.3)
p:(1,1)→(i,j) 0 0(i ,j )∈p

where p : (1, 1) → (i, j) indicates an up-right paths connecting cells (1, 1) and (i, j). The
quantity in the right hand side of (1.3) is known as last passage percolation time in theenvironment Gi,j = {gi0 ,j 0 : 1 ≤ i0 ≤ i, 1 ≤ j 0 ≤ j} and it is an object with deep meaning in
combinatorics. The celebrated Robinson-Shensted-Knuth (RSK) correspondence [Sag01]
provides a bijection between matrices of integers Gi,j and pairs of semi-standard tableaux2(P, Q) of the same shape λ. A crucial feature of the RSK algorithm is that the shape λis determined by the last passage percolation times of Gi,j and in particular λ1 = T (i, j).
Assuming that random variables gi0 ,j 0 are geometrically distributed as Geo(xi0 yj 0 ) for some2
Given a Young diagram λ = (λ1 ≥ λ2 ≥ · · · ≥ λ` > 0) a semi standard Young tableaux P is a Nvalued filling of cells {(i, j) : 1 ≤ j ≤ λi } that is strictly increasing along columns and weakly increasingalong rows. Here λ is the shape of P .

9set of positive parameters xi0 , yj 0 , a simple calculation and a careful use of the RSK showthat
Prob{T (i, j) = k} ∝X

iY

Gi,j :T (i,j)=k α=1

RSK

=

X

λ:λ1 =k




P

xα

X

γ gα,γP :sh(P )=λ

j
Y

P

g

yβ δ δ,ββ=1



xP  X

P :sh(Q)=λ



yQ =

X

(1.4)
sλ (x)sλ (y),λ:λ1 =k

Q #{c:P (c)=i}where the monomial xP = i xi
and normalization factor in the right hand side
Qi Qj
is α=1 β=1 (1 − xα yβ ). Functions sλ , defined by the sum over tableaux in the parentheses are the Schur symmetric polynomials and are fundamental objects in representation
theory and algebraic combinatorics. They are basis of the space of symmetric functions
and they possess a large number of properties and symmetries, including determinantal representations and summation identities. Their emergence in the study of corner
growth model is evidence of its integrability. In particular (1.4) shows that the study ofan abstract probability measure on Young diagrams weighted by Schur polynomials
S(λ) ∝ sλ (x)sλ (y),(1.5)

gives informations about a meaningful physical system once we consider the marginal
projection S(λ : λ1 = k). The probability measure S(λ) takes the name of Schur measure
and was introduced in literature by Okounkov in 1999 [Oko01]. Among all its outstanding
properties, the Schur measure is a determinantal point process and in fact it was employedby Okounkov in the study of a system of free fermions.
The connection with Schur measures is not the only way to ”solve” the corner growthmodel. Other techniques involve connections with random matrices [Joh00], GelfandTsetlin pattern representations [Sas05] and many more; see [Sep09] for a review. In
particular an extensive study of this model led to the complete characterization of the
KPZ fixed point [MQR17].
There also exists several other models that can be described in terms of Schur measures.
These include exclusion processes [Joh00], polynuclear growth models [PS02; Fer04], random polymers at zero temperature in various geometries [BZ19]. Moreover a measure of
the form (1.5) admits straightforward generalizations replacing Schur polynomials withmore general symmetric functions, e.g. Macdonald polynomials [Mac95]. We will review
such extensions below in section 1.1.3 and construct a new class of measures, based on
spin q-Whittaker polynomials, in chapter 3.
Schur processes and TASEPs
The mapping contained in computation (1.4) is certainly not the only way to make physicalsense of the Schur measure. A different idea, that we will largely explore in this thesis,
allows to relate the Schur measure to Markov processes using certain summation identities.

10

To explain this we need to introduce skew variants of Schur polynomials. The skew Schurpolynomials sλ/µ [Mac95] can be defined as the coefficients of the branching expansionXsλ (x1 , . . . , xn ) =sµ (x1 , . . . , xn−1 )sλ/µ (xn ).
(1.6)µ

They are labeled by pairs of interlacing Young diagrams µ, λ, that is
λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · ·(1.7)

and most importantly they satisfy the skew Cauchy identity
X
1 X
sλ/κ (y)sµ/κ (x) =sν/λ (x)sν/µ (y).
1 − xy κν(1.8)

Relation (1.8) is essential for probabilistic interpretation of skew polynomials and suggests
the construction of transition probabilities bringing the configuration κ ⊆ µ, λ 7→ µ, λ ⊆ νand viceversa. One example (but not the only one) of such transition probabilities canbe obtained taking the independent coupling as
Ufwd
x,y (κ → ν|λ, µ) ∝ sν/λ (x)sν/µ (y),Ubwdx,y (ν → κ|λ, µ) ∝ sλ/κ (y)sµ/κ (x)

∅

ν

λ
Ufwd

κ

∅

Ubwd

∅

µ
∅

(1.9)

λ(1,3)

λ(1,2)λ(2,2)

λ(1,1)

λ(2,1)

λ(3,1)∅∅∅

Figure 1.3: Left: Forward and backward transition operators. Right: Construction of arandom field using Ufwd , where lighter arrows correspond to moves happening later in the
update.
Performing moves weighted by Ufwd we can construct recursively a full field of Youngdiagrams {λ(i,j) : i, j ≥ 0}, starting from initial conditions λ(0,j) = λ(i,0) = ∅ as done in
fig. 1.3. We see that such random field describes a growth process of Young diagrams andas a result of branching rules its joint probability laws will be written in terms of productsof skew Schur polynomials. In particular the single point law is the Schur measureProb{λ(i,j) = λ} ∝ sλ (x1 , . . . , xi )sλ (y1 , . . . , yj ).
11(1.10)

and the joint law across a straight segment is the Schur process of Okounkov and Reshetikin[OR03]
Prob{λ(1,j) = µ(1) , . . . , λ(i,j) = µ(i) } ∝ sµ(1) (x1 )sµ(2) /µ(1) (x2 ) · · · sµ(i) /µ(i−1) (xi )sµ(i) (y1 , . . . , yj ).
(1.11)
We can take a look at fig. 1.4 left panel to visualize a schematic example of a sequence ofinterlacing Young diagrams such as those weighted by the Schur process.
Random fields built following this recipe are certainly integrable, since nice properties of Schur polynomials allow exact computations for a multitude of observables: they
are determinantal point processes [OR03] as a result of Jacobi-Trudi formulas, their correlations are given by integral formulas thanks to certain eigenrelation [Agg15] and soon. Nevertheless this abstract algebraic construction does not reveal much about thetrue physical meaning of the process. That is we do not know a priori if any reasonable
stochastic process will be described by a measure of the form (1.11). This is a problem
that needs to be addressed separately by carefully studying Markov maps Ufwd , Ubwd . For
the case of fields built from Schur polynomials this sampling problem admits a numberof different solutions, from which one deduces that marginals of the Schur process are
sampled by very natural stochastic systems: the Totally Asymmetric Simple Exclusion
Process (TASEP) and its push variant (pushTASEP); see fig. 1.4. This last result was
found first by Borodin and Ferrari in [BF14], where this general construction was also introduced. Employing the RSK algorithm more refined constructions of Markov operator
Ufwd lead to analogous results with TASEP and pushTASEP replaced by their discrete
time analogs [MP17]. In this thesis, we will construct generalizations of these argumentsin the context of growth processes of Young diagrams weighted by other special function,
namely the spin Hall-Littlewood and the spin q-Whittaker functions; see chapter 2 andchapter 3.
We close this short section stressing the fact that probability measures on interlacing
arrays are not an unusual object. In fact they are rather common in random matrix theory,
where eigenvalues of minors of symmetric and hermitian matrices arrange themselves ininterlacing sequences3 . Alternatively one can recall that Dyson’s Brownian motion [Dys62]
for GUE random matrices arises from ensembles of reflecting Brownian particles, condition
that can be seen as a continuous analog of the interlacing. One can therefore think of the
Schur process (1.11) as a discrete versions of these common constructions.

1.1.3

Beyond Schur processes

The blueprint described above to construct growth processes on Young diagrams startingfor skew Cauchy identities (1.8) works in much higher generality. A truly successful
application of this scheme comes replacing the Schur polynomials sλ/µ with Macdonald
polynomials Pλ/µ [Mac95]. The Macdonald processes are introduced by Borodin andCorwin in 2011 [BC14] and they sit on top of a vast hierarchy of integrable stochastic
systems; see fig. 1.5. They simultaneously generalize particle processes, random polymers
and last passage percolation models, ensembles of random matrices and also the KPZ
3this is the Cauchy interlacing theorem

12

(4)

(4)

µ4

(4)

µ3
(3)

(3)

µ3

(2)

µ1

(TASEP)

(3)

µ2

µ2

rate 1

(4)

µ2

µ1

rate 1(2)µ1(pushTASEP)

(1)
µ1

push

Figure 1.4: In the left panel we see an interlacing array µ(1) ≺ · · · ≺ µ(4) . Under the(i)(i)
Schur process, coordinates µ1 (circled in blue) and µi (circled in red) form autonomous
Markov processes. They are equivalent to pushTASEP and TASEP, whose dynamic issketched in the right panel. In the TASEP particles jump to their right neighboring sitewith rate 1 provided the site is empty. In the push-TASEP, in case the target site is
occupied a deterministic push happens.
stochastic partial differential equation. Algebraic structures of Macdonald polynomials
descending from representation theory survive in each of these systems and provide tools toanalyze exactly a range of observables. In particular the theory of Macdonald processes
offers a way to rigorously solve the KPZ equation under a class of initial conditions[BC14; Bor+15a; Bar+18]. Other techniques to solve the KPZ equation include duality
arguments, moment expansion, Bethe Ansatz and so on; see [ACQ11; SS10; CLR10;
Dot11; LC12; IS13].
In the case of Macdonald processes the problem of characterizing interesting marginal
processes, like the corner growth model or the TASEPs in the Schur case, has attracted
considerable attention in the last decade. A certain simplification of the general Macdonald processes, called q-Whittaker processes, was found to be sampled by randomized
versions of the RSK algorithm [OP13; BP16; MP17], relating marginals to q deformationsof TASEPs of fig. 1.4. Other uses of the RSK algorithm in Macdonald processes givealso rise to the stochastic six vertex model [BM18] and to random polymers [Cor+14],[OSZ14], [OO15], [CSS15].
In parallel to developments strictly pertaining Macdonald processes in the last 10
years, different techniques and theories have emerged and attracted their own share ofattention. The most remarkable of these is arguably related to Yang-Baxter integrability.
The earliest and most fundamental example of such systems is the stochastic six vertex
model dating back to 1992 [GS92] that was studied by Gwa and Spohn to confirm kinetic
roughening exponent of KPZ models. More recently the stochastic six vertex model was
resurrected [BCG16] and largely generalized using algebraic techniques of representation
theoretic origin. This is the higher spin vertex model [CP16], which we will discuss morein detail in the next section. The higher spin vertex model is a rather general stochastic
integrable system that possesses as particular case a number of other interesting models:these are exclusion processes like the q-Hahn TASEP [Pov13] and systems of random
walkers in random environment like the Beta polymer model [BC16a].
Before the works [BMP19; MP20] reported in this thesis it was not known if a description of the higher spin vertex model in terms of dynamics of interlacing arrays was13

Spin Hall-Littlewood (t, s)
Dynamic stochastic six vertex model; stochastic higher
spin six vertex model

Macdonald (q, t)
q=0

t=0

Spin q-Whittaker (q, s)Stochastic higher spin six vertex
model; q-Hahn TASEP; 4 φ3 modeland q-Hahn PushTASEP

s=0

q→1

s=0

Hall-Littlewood (t)Stochastic six vertex model; ASEP
t=0

q-Whittaker (q)q-TASEPs; q-PushTASEPs

Spin Whittaker (S)Beta polymer models

q→1

q=0

S → +∞

Whittaker
Log-gamma polymer;
O’Connell-Yor polymer

Schur
Directed last / firstpassage percolation

Figure 1.5: A scheme of various families of symmetric functions together with stochastic
systems based on them. Arrows indicate degenerations or scaling limits. The two families
which are our main focus are indicated in bold.
possible or not. Among the results we will see in chapter 3 we find a positive answer tothis problem employing random fields with spin q-Whittaker symmetric functions [BW17;
MP20]; see fig. 1.5. Consequences of this finding are algebraic derivations of formulas forobservables of the higher spin vertex model or also more generally the possibility to employ
line ensembles techniques as the one introduced in [CH14] for to study general stochastic
integrable models. These will be object of future studies.

1.1.4

The higher spin vertex model

Let us now informally define the main character of this thesis, whose investigation drives
the majority of results we present. The higher spin vertex model was introduced by
Corwin and Petrov in 2015 [CP16] in an attempt to unify under the same umbrella thegrowing collection of different stochastic integrable models that had been studied in theprevious years in the context of KPZ universal phenomena. These include a variety of
exclusion processes, zero range processes and polymer models in continuous and discretetime.
The higher spin vertex model is a probability measure on ensembles of directed pathsin the quadrant Z≥0 × Z≥0 ; see fig. 1.6, left. Paths emanate from boundary vertices andpropagate through the lattice keeping up-right direction, where the propagation rules are
given by vertex weights Lur
x,v of fig. 1.7. More precisely the process is constructed as follows:• each vertex (0, j) emanates bvj paths initially directed to the right;• each vertex (i, 0) emanates bhi paths initially directed upwards;• the probability of observing a configuration (α1 , β1 ; α2 , β2 ) at vertex (i, j), conditioned on the configuration at all vertices (i0 , j 0 ) with i0 + j 0 < i + j, is given by
14

j
.
.
.
2
1
0

3

2

2

0

0

..
.

2

2

2

0

0

3

2

1

1

0

0

1

0

0

0

−1

0

0

−1 −1 −1

0

1

2

···

2
1
i+j =4
1

i

2

3

···

Figure 1.6: Left: a configuration of up-right paths. The numbers in blue are the valuesof the height function HHS at each vertex.
Right: a configuration of paths in the triangle i + j ≤ 4. The probability to observe thisspecific arrangement is reported in (1.13).
Lurxi ,vj (α1 , β1 ; α2 , β2 ). Moreover, this event is independent of choosing arrow configurations at other vertices . . . , (i − 1, j + 1), (i + 1, j − 1), . . . on the same diagonal.
Informally the probability of a configuration of paths C in the full lattice is given by
Prob(C) =Yi≥1

Prob(bhi ) ×Yj≥1

Prob(bvj ) ×

Y

i,j≥1

Lur
xi ,vjvertex config.
.
at (i, j)(1.12)

We notice that the system is completely inhomogeneous both in the i and j direction
and this is due to the fact that weights Lur
xi ,vj depend on sets of parameters xi , vj otherthan on global parameters s, q. We also will always assume that boundary paths B h =(bh1 , bh2 , . . . ), B v = (bv1 , bv2 , . . . ) are sampled independently from each other and from thebulk configuration. Since vertex weights L are assigned the measure on the path ensemble
is determined only assigning boundary conditions B h , B v and we denote it with P(B h , B v ).
To illustrate more clearly how the construction woks, we report here the explicit
expression of the probability of observing, in the region {(i, j) ∈ Z≥0 × Z≥0 : i + j ≤ 4},
the configuration drawn in fig. 1.6 right panel. We haveProb(fig. 1.6 right) =Prob(bh1 = 2, bh2 = 1, bh1 = 0) × Prob(bv1 = 0, bv2 = 1, bv1 = 0)






ur
ur
× Lur
L
L
x1 ,v1x2 ,v1x3 ,v1(1.13)

 



× Lur
Lur
Lur
.
x1 ,v2x2 ,v2
x1 ,v3

An equivalent description of the ensemble can be given defining a height function HHS .
Given a configuration of paths and a vertex (i, j) we set

HHS (i, j) = #{occupations at horizontal edges} − #{occupations at vertical edges},(1.14)
15

α2

g

g−1

g+1

g

α1

g

g

g

g

Lur
x,v (α1 , β1 ; α2 , β2 )1+q g xv
1+xv

xv(1−q g )1+xv

1+q g sx
1+xv

xv−q g sx
1+xv

β1

β2

Figure 1.7: The stochastic vertex weights Lur
x,v for the up-right stochastic higher spin six
vertex model. Configurations that are not of the type reported in the table are assigned
weight zero. The stochasticity condition means that fixed down and left edges occupation
numbers α1 , β1 , the sum of weights Lur
x,v (α1 , β1 ; α2 , β2 ) over all possible α2 , β2 is 1.
counted on the union of two segments ( 12 , 12 ) → (i + 21 , 12 ) → (i + 12 , j + 12 ); see fig. 1.6 for an
example. The height function allows to relate the higher spin vertex model to exclusion
processes interpreting the difference H(n, t) − H(n + 1, t) as the gap between consecutive
particles and hence weights Lur as jump rules.
The stationary higher spin vertex model has been studied in [IMS20]. The modelis potentially fully inhomogeneous as weights Lur can depend on both coordinates i, j,therefore a proper notion of translation invariant measures ought to be defined. Forh= (bhi+1 , bhi+2 , . . . ),this consider random boundary conditions B h , B v and their shifts B+iv= (bvj+1 , bvj+2 , . . . ). We say that a measure P(B h , B v ) is translation invariant if theB+jrestriction of the ensemble to the quadrant Z≥i × Z≥j originating from a generic pointhv(i, j) is equivalent in distribution to P(B+i, B+j).
Translation invariant measures for the higher spin vertex model form a one parameterfamily. They are obtained fixing the law of boundary conditions as


d vj
v
bj ∼ Ber,bhi ∼ qNB(−sxi , d /xi ),(1.15)
1 + d vj
where Ber indicates the Bernoulli distribution and qNB is the q-negative binomial distribution defined by
X ∼ qNB(b, p)=⇒

P(X = n) = pn(b; q)n (p; q)∞,(q; q)n (pb; q)∞for all n ∈ Z≥0 . (1.16)

We used the notion of q-Pochhammer symbol (a; q)n (A.1). We refer to the model withboundary conditions (1.15) as the stationary higher spin vertex model with density d . We
will study its asymptotic properties and establish Baik-Rains characteristic fluctuations
in chapter 4 based on results of [IMS20].

1.1.5

Spin Hall-Littlewood functions

The salient feature that makes the higher spin vertex model an integrable systems is thec2 ) R-matrices and they satisfy the Yangfact that stochastic weights Lx,v are in fact Uq (slBaxter equation. This equips the model with a large variety of symmetries that produce
16striking implications and whose investigation is at the moment still active; see [Pet19;
BGW19] for very recent developments.
Coordinate Bethe Ansatz eigenfunctions of the transfer matrix of the higher spinvertex model are known as spin Hall-Littlewood functions. They were defined in [Bor17]
and their main combinatorial and algebraic properties were studied in [Bor+15c; BP18a].
These include the symmetrization formula [BP18a] `(λ)
 Y
n λi 
ui − quj Y Y
ui − sj Y ui
(1 − q)n Xσ,Fλ (u1 , . . . , un ; s) =(q; q)n−`(λ) σ∈Sui − uj i=1 j=1 1 − sj ui i=1 ui − sλi
1≤i<j≤nn(1.17)
that we largely make use of in chapter 2. Specializing all spin parameters sj = 0, functions
Fλ reduce to the Hall-Littlewood polynomials and additionally setting q = 0 they become
the Schur polynomials. Most remarkably spin Hall-Littlewood functions form a complete
set as a result of orthogonality relations both in the λ and in the u variables.
In [BMP19; MP20] eigenrelations for them with respect to Macdonald type operators
were first established. We will review these results in chapter 2.

1.2

Results

Before we enumerate the list of results presented in this thesis we like to dedicate a small
subsection to stress motivations driving the author in the extensive study of stochastic
integrable systems.

1.2.1

Motivations

In the last twenty years the study of stochastic integrable systems has brought significant
advances to the understanding of properties and boundaries of KPZ universal phenomena.
In the last decade in particular the understanding of algebraic and combinatorial structures underlying certain models of growth has allowed rigorous derivations of solutions of
the KPZ equation [ACQ11; SS10] along with a full descriptions of conjectural universal
limiting objects [MQR17].
Among the highest goals of the theory is to capture a general set of conditions thatwould automatically provide a physical process of KPZ behaviors and hence to establish a“central limit theorem” for strongly interacting systems. Nevertheless at the current statesuch general type of result appears to be out of hand: rigorous statements are proven only
for a few toy models and the real breadth of the universality class is still object of active
investigations.
We are then forced to develop a solid theory of the few models we are able to solve,
possibly discovering new ones with the aim of expanding the scope of their associateduniversality. In doing so, in the last decade a number of models were defined. Startingfrom the classical Totally Asymmetric Simple Exclusion Process (TASEP) which we briefly
touched upon in section 1.1.2, it was realized that its exact solvability does not breakby adding additional parameters in specific ways. ASEP and q-TASEP are two such17

deformations (see also the q-boson of [SW98]): both systems depend, albeit in a different
ways, on a quantization parameter q, governing the asymmetry of the transport. By
taking scaling limits in q one discovers that integrability of discrete systems translates into
solutions of otherwise intractable continuous models, like directed polymers in random
media [OY01] or the KPZ equation.
More recently it was found that integrability survives the addition of a further parameter, which in this thesis we denote with s, leading to the introduction of the q-Hahn
TASEP [Pov13] and of the higher spin vertex model [CP16], defined in section 1.1.4. The
presence of two “quantization” parameters offers the possibility of richer scaling limits,that in the particular case of the q-Hahn TASEP led to the remarkable discovery thatcertain phenomena in diffusion in random media or certain statistics in extreme value
theory of strongly correlated processes obey KPZ universal laws [BC16a].
The scope of this thesis is that of deepening the understanding of the stochastic integrable systems in the (1+1)-KPZ universality class. We do so by defining a general
integrable system, which we call spin q-Whittaker process. The importance of this newmodel lays in the fact that it simultaneously generalizes all the models introduced above
and a number of other ones, which we will discuss below. This solves a question that was
asked several times in literature (eg. [Cor14]) and that is to inscribe integrable stochastic
models in more general dynamics on interlacing arrays in the spirit of the Macdonald
Processes [BC14]. Properties of the spin q-Whittaker processes are described by a familyof special functions we study extensively: the spin q-Whittaker polynomials. Based on an
idea of [BC14], we also propose a systematic scheme to recover formulas for observablesof the various systems, that passes through the computation of eigenrelations for the
spin q-Whittaker polynomials. This has also allowed to solve certain conjectures which
had appeared in literature in recent years [CMP19] regarding the rigorous derivation ofobservables for models with diverging moments. We imagine that the simplicity of thisprocedure will find applications in establishing refined convergence results of the different
systems to the universal object (eg. convergence of two-point distribution as in [Dim20]).
From a more probabilistic standpoint one of the consequences of our constructions is that
there exist Gibbsian line ensembles behind all of the integrable vertex models. These
objects, considered first in [CH14], represent a powerful tool to show regularity of convergence of various models to universal objects and their analysis will be object of future
works.
Along the way we study the stationary states of the higher spin vertex model, as a
general model of KPZ growth, and establish the convergence of the one point function to
the so-called Baik-Rains universal law. A brief discussion on experimental relevance of
this last result is offered below.
We are now ready to explain in detail the results contained in this thesis.

1.2.2

Improving the theory of spin q-Whittaker polynomials

In [BW17], Borodin and Wheeler defined a family of symmetric functions that they called
spin q-Whittaker polynomials. Their basic properties were studied from a combinatorial18

standpoint leveraging Yang-Baxter relations, but no result about their relations to probabilistic integrable systems or about their relevant algebraic properties was given. Our
first stream of results is an attempt to resolve this issue.
We introduce modified versions of the spin q-Whittaker symmetric polynomials. Their
expression can be given through a usual branching expansionFν (x1 , . . . , xN ) =XnYFν i /ν i−1 (xi ),(1.18)∅=ν 0 ≺ν 1 ≺···ν n =λ i=1

where ν i are non-negative signatures of i parts, ≺ is the interlacing condition (1.7) andthe branching coefficients arekY(−s/x; q)λi −µi (−sx; q)µi −λi+1 (q; q)λi −λi+1

λ = λ1 ≥ · · · ≥ λk+1 ≥ 0µ = µ1 ≥ · · · ≥ µk ≥ 0.
i=1
(1.19)We used again the notion of q-Pochhammer coefficient (a; q)n (A.1). Our polynomials areslightly more general than the Borodin–Wheeler’s version FBW[BW17], since we haveλFλ/µ (x) := x|λ|−|µ|(q; q)λi −µi (q; q)µi −λi+i (s2 ; q)λi −λi+i,Fλ (x1 , x2 , . . . , xn ) x1 =0 = FBW
λ (x2 , . . . , xn ),but, rather surprisingly, this little improvement grants them of a number of additional
properties, including shifting, summation identities and eigenrelations. We outline now alist of their features all of which are proven in chapter 2.
• The Fλ are symmetric with respect to exchange of the x variables. This property
follows from a combination of two Yang-Baxter type relations. One is a classic2 ) R-matrix [Man14]. The other is a
cal Yang-Baxter equation for the fused Uq (slc2 ) R-matrix and a new ”corner” weight we
new commutation relation between Uq (slintroduce; see fig. B.7 and proposition B.2.1.
• Under the degeneration s = 0, both families Fλ and FBWcoincide and turn into theλusual q-Whittaker polynomials. The different scaling s → 0, xs = const. produces
the Grothendieck polynomials, a key object in algebraic combinatorics. This second
fact was also not observed before in literature.
c2 )• They satisfy skew Cauchy identities as a result of Yang-Baxter equations for Uq (slR-matrices and corner weights.

• We present q-difference operators D1 , D1 which act on our new spin q-Whittakerpolynomials diagonally as
D1 Fλ = q λN Fλand

D1 Fλ = q −λ1 Fλ .

These represent one parameter generalizations of the first q-Whittaker operators(these are t = 0 degenerations of the Macdonald operators from [Mac95]). The
emergence of a theory of operators is a strong hint of algebraic relevance of our spin
q-Whittaker polynomials.
19• We observe that the operators D1 , D1 can be represented as conjugations of the first
q-Whittaker q-difference operators. From higher q-Whittaker operators we thus gethigher order q-difference operators commuting with either D1 or D1 (the conjugations leading to D1 and D1 are different, even though these operators commute).
The higher order operators coming from the q-Whittaker operators are not diagonalin the spin q-Whittaker polynomials.
Our modification of the spin q-Whittaker polynomials originates from computer experiments informed by the existing definition from [BW17] combined with the desire tohave q-difference eigenoperators. A particular case of one of the eigenoperators appearedin [BMP19], while the D1 , D1 we present here were introduced in [MP20]. This led us
to discover the existence of commutation relations involving corner vertex weights. It
would be very interesting to connect our corner vertex weights and the corresponding
Yang–Baxter equation with known integrable vertex model constructions.
The major unsolved problem in the theory of spin q-Whittaker polynomials, that
we believe could uncover their true algebraic meaning, is represented by the conjectural
orthogonality relationZ?
Fλ (z1 , . . . , zN ) Fµ (1/z1 , . . . , 1/zN ) mN
(1.20)q,s (z1 , . . . , zN ) dz1 · · · dzN = cλ 1λ=µ ;TNsee section 2.7 for precise definitions of quantities. This represents a natural generalizationof the torus scalar product for Macdonald polynomials [Mac95]. In the Macdonald casethe existence of families of commuting eigenoperators suffices to show the orthogonality,
but in our case operators could only suggest us the exact form of the scalar product andare not enough to conclude. Our conjecture was tested numerically for small examples,a more thorough check was made under the assumption q = 0 and in the text we also
provide a few arguments in support of it.

1.2.3

Characterizing marginals of random fields

We provide a detailed anaysis of random fields of Young diagrams weighted by spin
Hall-Littlewood and spin q-Whittaker functions. Leveraging on their combinatorial andalgebraic properties we identify two classes of Markov marginal projections and match
their distribution with a number of general integrable stochastic systems. The theoryof eigenrelations offers a systematic way to study multi point joint distribution of theprocesses, that in turn allows us to recover determinantal formulas for the one-point
distrubutions of certain observables using a strategy introduced in [BC14]. The following
is a list of main results contained in chapter 3.
• In section 3.1 we review a general scheme to construct integrable probability measures on fields of Young diagrams λ = {λ(i,j) : i, j ≥ 1} from pairing of symmetricfunctions satisfying skew Cauchy identities. In the case of spin Hall-Littlewood(sHL) and spin q-Whittaker (sqW) functions, leveraging on Yang-Baxter relations,
we build up Markov processes with the remarkable property sampling the respective
20

stochastic 6VM

up-right HSVM

up-left HSVM

q-Hahn TASEP

4 φ3 vertex model

Figure 1.8: Schematic representation of marginals of fields studied in section 3.3. The
first row of a sHL/sHL field is equivalent to the stochastic six vertex model (green). The
first and the last row of a sqW/sHL field are equivalent respectively to the up-left andup-right directed higher spin vertex model (blue). Finally the first and the last row ofa sqW/sHL field are equivalent respectively to the 4 φ3 vertex model and to the q-Hahn
TASEP (red).
fields. The construction of these Yang-Baxter fields uses a technique called bijectivization that was introduced in [BP19]. As a result of bijectivization certain classes
of projections of the fields are found to be form autonomous Markov processes.
c2 ) Yang-Baxter equation reduces, in the limit q → 0 to
• Bijectivization of the Uq (sllocal moves of the Robinson-Schensted-Knuth insertion algorithm. This reduction
to the RSK is presented in section 3.2.
• We define the sHL/sHL field λ, where joint law of partitions is given by productsof skew sHL functions. For example the single point distribution isProb(λ(i,j) = λ) ∝ Fλ0 (u1 , . . . , ui )F∗λ0 (v1 , . . . , vj ).

(1.21)(i,j)

The first row marginal of the field λ1 forms an autonomous Markov process and it
is equivalent in distribution with the height function of a stochastic six vertex model;
see fig. 1.8. This result was obtained first in [BBW18; BP19]. Additionally, making
use of scaled geometric specializations of sHL functions studied in section 2.5 wecan produce fields of Young diagrams with gibbsian random boundary conditions.
(i,j)
These, at the level of the λ1 marginal produce the six vertex model with two sided
Bernoulli boundary conditions studied in [Agg18].
• In the sqW/sHL field λ joint law of signatures is given by product of pairs of skew
sqW and sHL functions; for example the single point distribution isProb(λ(i,j) = λ) ∝ Fλ (x1 , . . . , xi )F•λ0 (v1 , . . . , vj ).

(1.22)

Using bijectivizations we characterize two autonomous Markov marginal projections.
(i,j)
The first row marginal λ1 is equivalent to the height function of a variant of thehigher spin vertex model with up-left directed paths. On the other hand the last21

(i,j)

row marginal λi is matched with the height function of a higher spin vertex modelwith up-right directed paths; see fig. 1.8. These results were obtained in [BMP19;
MP20].
• We finally construct the sqW/sqW field λ and in it joint law of signatures is given
by product of pairs of skew sqW and sqW functions; for example the single point
distribution isProb(λ(i,j) = λ) ∝ Fλ (x1 , . . . , xi )F∗λ (y1 , . . . , yj ).

(1.23)

The fields possess two natural Markov marginal projections. The first row marginal(i,j)
λ1 is equivalent to the height function of a general exclusion process with pushingmechanism called 4 φ3 vertex model with up-left directed path. This was introduced(i,j)
in [BMP19; CMP19]. The last row marginal λi is matched with the position ofa tagged particle in a q-Hahn TASEP; see fig. 1.8. These results were obtained in[BMP19; MP20].
• In a simplified “Plancherel” (or “Poisson-type”) continuous time limit we construct a
Markov dynamics on interlacing arrays under which the last rows marginally evolve
as a continuous time version of the q-Hahn TASEP (appeared in [BC16b]). Our
new two-dimensional continuous time dynamics is a model of 2d anysotropic growth
and represents a one-parameter deformation of the q-Whittaker 2d-growth model
introduced in [BC14, Definition 3.3.3]. The latter growth model has continuous time
q-TASEP as the last row marginal dynamics.
• Eigenrelations for sHL and sqW functions provide nested integral formulas for joint
q-moments of the fields. In particular the generating function of single point q(i,j)(i,j)
moments of coordinates λ1 or λi can be rearranged in a Fredholm determinant.
In the simple case of the sHL/sHL field this gives a determinatal formula for theheight function H6V of six vertex model with domain wall boundary conditions. We
have


1
= det (Id + K)L2 (C) ,ζ ∈ C \ R>0 ,(1.24)
E(ζq H6V (i,j) ; q)∞where the integral kernel K is given byQi(w − uk )withf(w) = Qk=1.
j
k=1 (w − vk )(1.25)
Formulas for models with random boundary conditions are also accessible thanks
to analytic continuations and fusion techniques.
1
K(w, w ) =2i0Z(−ζ)r f(w)/f(q r w)dr,r0d+iR sin(πr) q w − w

The main immediate consequence of the connection between random growth processesof Young diagrams weighted by our symmetric functions and stochastic integrable systems
in the KPZ universality class is a fully algebraic unified theory of observables for thesemodels. This was used in [BMP19] to prove conjectures about determinantal expressions
22

for certain systems of particles and polymers formulated in [CMP19]. More in general
our theory brings the realm of complicated integrable systems, like the higher spin vertex
model, closer to that of determinantal point processes like the Schur processes. We envision that further studies might reveal additional connections extending our theories tothe evaluations of more interesting observables, including multi point joint distributions.
A possible strategy to describe multi point statistics would be to follow techniques introduced in the very recent work of Dimitrov [Dim20], where he characterized the two point
correlations of the stochastic six vertex model using difference operators analogous to ourD1 . We leave these investigations for future works.

1.2.4

Asymptotic analysis for the stationary higher spin vertex
model

In chapter 4 we establish KPZ fluctuations for the stationary higher spin vertex model.
Our starting point is a Fredholm determinant expression for the q-Laplace transformof the height function that was obtained in [IMS20]. The derivation of the formula weuse employs an elliptic generalization of the Cauchy determinant, following techniques
invented by Imamura and Sasamoto in [IS19].
Stationary growth processes in the KPZ class exhibit typical fluctuations and characteristic 1:2:3 scaling only if we follow the evolution of the interface along a specificdirection. To explain this phenomenon consider the KPZ equation. In the stationary evolution the height profile is a Brownian motions and we consider its displacement between
two point in space and time that with no loss of generality we take as (0, 0) and (x, t).
Such height displacement possesses in the long space time limit two different regimes.
Around the characteristic line of the Burgers equation associated with the dynamics, that
one can understand as the direction of the growth, the height scales according to KPZ
exponents and its one point distribution is governed by the Baik-Rains distribution F0(see Definition 4.2.3). Alternatively, when we move away from the characteristic line
the size of fluctuations coming from the stationary initial data, the brownian motion,
overwhelms any possible nontrivial behavior produced by the random dynamics and theheight performs a gaussian process. The same is true for discrete models where insteadof the brownian motion, the stationary profile of the height is given by a sequence ofindependent increments.
For the stationary higher spin vertex model we now want to give the exact expressionof scaling parameters defining the characteristic line and the expected behavior of theheight function H. For simplicity in this case we suppress time inhomogeneities and we
set vj = v. We make use of q-polygamma type functions υk defined in chapter A. For
non-negative integers k consider the functionsnak (d ) = υk (qvd ) − υk (vd ),hk (d ) =23

1X
(υk (d /xi ) − υk (−d s))n i=2(1.26)

and, depending on the parameter d , define the quantities1/3
1h1 (d ),
η0 = κ0 a0 (d ) − h0 (d ),γ=−(κ0 a2 (d ) − h2 (d )).
κ0 =a1 (d )
2

(1.27)

We assume that the functions hk always converge in the large n limit and we refer to thecurve (n, κ0 n) as the characteristic line of the stationary higher spin vertex model. For
random growth models usually the characteristic line is expressed as a function of thetime t, rather than of the coordinate n, but in our case, since the system exhibits spatial
inhomogeneities we find more natural to adopt the notation (n, κ0 n). The parameter η0multiplied by n is readily understood as the expectation E(H(n, κ0 n)), whereas γ will be
used to describe the size of the characteristic fluctuations of H around η0 . By slightly
perturbing quantities κ0 , η0 we can analyze the asymptotic behavior of H in a region ofsize n2/3 around the characteristic line. For this we extend the definitions given in above
settingh2 a1 − h1 a2 $(1.28)
κ$ = κ0 +a21
γn1/3
η$ = η0 +a0 (h2 a1 − h1 a2 ) $h2 a1 − h1 a2 $2
+
,
a21
γn1/3
a1
γ 2 n2/3(1.29)

where $ is a real number parameterizing the displacement from the characteristic line
and functions ak , hk are evaluated at d . We come now to state our result.
Theorem 1.2.1. Consider the stationary higher spin vertex model with density d . Then,for any real numbers $, r we haveH(n, κ$ n) − η$ n
> −r = F$ (r),(1.30)
lim P
n→∞
γn1/3
where F$ (r) is the Baik-Rains distribution presented in Definition 4.2.3.
For a more precise statement of this theorem we refer to chapter 4.
An alternative procedure we could have followed to establish Baik-Rains fluctuations
for the height functions of the higher spin vertex model is by using Fredholm determinant
formulas derived in chapter 3. These turn out to be different than those obtained throughelliptic determinant calculations and their asymptotic analysis would have been morealong the line of [Agg18] where Aggarwal studied the stationary six vertex model.

1.2.5

A new class of functions: the spin Whittaker functions

Our final series of results deals with the q → 1 scaling limit of the spin q-Whittakerpolynomials and of stochastic systems associated with them. This leads to the definitionof a new class of functions that can be seen as one parameter deformation of Whittaker
functions [Jac67; Kos78]: we call them the spin-Whittaker functions.
The spin Whittaker functions fX1 ,...,XN (L1 , . . . , LN ) depend on a global parameter S >0, complex numbers X1 , . . . XN , such that |Xi | < S and a continuous signature LN =24

(L1 ≥ . . . ≥ LN ≥ 0). They may be defined via a recursive Givental-type integral
representation. Let L0N −1 and LN be interlacing sequences:
1 ≤ LN ≤ L0N −1 ≤ LN −1 ≤ . . . ≤ L01 ≤ L1

(notation: L0N −1 ≺ LN ) and define the branching functions
−X
1
LN · · · L1
0fX (LN −1 ; LN ) :=(B(S + X, S − X))N −1 L0N −1 · · · L01

S+X−1 
1−2S
N
−1
Y
L0j S−X−1
Lj+1
Lj+1
×
1−
1− 0
1−
,
Lj
Lj
Lj
j=1

(1.31)

1, and, inductively,
where B(S + X, S − X) is the Beta function. Set fX1 (L1 ) := L−X
1ZdL
fX1 ,...,XN (LN ) :=fX1 ,...,XN −1 (L0N −1 ) fXN (L0N −1 ; LN ) 0 N −1 .
(1.32)LN −1
L0N −1 : L0N −1 ≺LN

To familiarize the reader with the notion of spin Whittaker functions we report thesimplest nontrivial case N = 2 that takes the simple formz
S + X, S + Y
S −X−Y,1 ≤ u ≤ z,
fX,Y (u, z) = (z/u) u
1−
2 F1
2S,
–
u
where 2 F1 is the Gauss hypergeometric function (A.24).
We now give a list of results concerning spin Whittaker functions contained in chapter 5.
• After establishing regularity and well posedness of the Givental representation (1.32),
we define a notion of dual spin Whittaker functions gY1 ,...,YM (LN ). The pair of functions f, g satisfies Cauchy identities an example of which reads
!
Z
M
N
Y
dLN
Γ(X1 + Yj ) Y Γ(Xi + Yj )Γ(2S)
fX1 ,...,XN (LN ) gY1 ,...,YM (LN ).
=LN
Γ(S + X1 ) i=2 Γ(S + Xi )Γ(S + Yj )j=1
(1.33)For the usual Whittaker functions, the first Cauchy type identity with M = N isdue to Bump and Stade [Bum89], [Sta02], [GLO08], and was later generalized in[Cor+14, (1.2)], [BC14, Section 4.2.1].

• Consider the spin q-Whittaker polynomial Fλ (x1 , . . . , xN ) and introduce the scalingq → 1,

xi = q Xi ,s = −q S ,q −λi = Li ,where S > 0, |Xi | < S, and 1 ≤ LN ≤ . . . ≤ L1 are fixed real numbers. We show in
Theorem 5.1.14 that under this scaling the convergenceFλ (x1 , . . . , xN ) −→ fX1 ,...,XN (L1 , . . . , LN ),(1.34)

holds uniformly on compacts. Similar results are also given for dual functions F∗
and g.
25

• We also define spin Whittaker processes. These are probability measures on interlacing sequences of continuous signatures L1 ≺ L2 ≺ . . . ≺ LN , with probability
weights expressed through the spin Whittaker functions. Cauchy type identity (1.33)
provides an explicit normalizing constant for the spin Whittaker process. We matchthe distribution of the marginal process L−1
k,k to the strict-weak beta polymer modelof [BC16a] (Theorem 5.2.6), and the distribution of the other marginal process Lk,1
to the other beta polymer like model which appeared in [CMP19] (Theorem 5.2.11).
• We show (Theorem 5.3.3) that the functions fX1 ,...,XN (LN ) are eigenfunctions of adeformation of the glN quantum Toda HamiltonianNX
1X 2
H2 := −∂ui +S −2(j−i) euj −ui (S − ∂ui )(S + ∂uj ).
2 i=1
1≤i<j≤N(1.35)

Introduce a change of variables Lj = S N +1−2j euj . Then in these variables we have!
N
1X 2
H2 fX1 ,...,XN = −X fX1 ,...,XN .
2 j=1 j• As S → +∞ and under the scaling Lj = S N +1−2j euj , Xj = −iλj , the operator
H2 and the spin Whittaker functions fX1 ,...,XN (LN ) formally reduce to the usual
quantum Toda hamiltonian Whittaker functions
!
N
N
−1
N
2
X
X
X
1
1
∂
H2Toda = −+eui+1 −ui ,H2Toda ψλ (uN ) = −λ2i ψλ (uN ).
2 i=1 ∂u2i
2
i=1
i=1
(1.36)
A similar reduction brings spin Whittaker processes to Whittaker processes from
[OCo12], [Cor+14], [BC14]. We do not fully justify these limit transitions, as thisrequires a much finer analysis and justification of the interchange of the S → +∞limit with Givental-type representations. However, we note that at the level ofmarginals, the strict-weak beta polymer becomes the strict-weak log-gamma polymer [BC16a, Remark 1.5]. We also show (Proposition 5.2.13) that the other beta
polymer type model turns into the log-gamma polymer from [Sep12].
The emergence of new classes of functions with a list of remarkable features is not a very
common event. In this thesis we define the spin Whittaker function in full generality andestablish some of they immediate properties. We do not consider the study of non trivial
scaling limits, that could reveal additional hidden structures. For example we do notinvestigate to what extent such functions arise from different combinatorial constructions,like the classical Whittaker functions and their relation to tropical RSK [OSZ14]. We will
explore these directions in further works.

26

1.2.6

Physical relevance of results

This thesis mainly focuses on mathematical properties of the stochastic integrable models
in the KPZ class. Nevertheless, our results clearly have relevance also from the experimental standpoint. Here we will briefly discuss two examples.
Transport in inhomogeneous space
The higher spin vertex model can be seen as a model of transport in inhomogeneous
space. To see this we naturally interpret the path ensemble as a collection of trajectories
of particles forced to move in one spatial dimension (the i coordinate in fig. 1.6 left) overtime (the j coordinate). A scaling of parameters v = ε ∼ 0, s = −1 transforms theprobability weights of fig. 1.7 into




g
ur
ur
∼ εxi (1 − q )Lxi ,ε∼ xi q g .
Lxi ,εggThis, following a time rescaling j = t/ε, translates the higher spin vertex model in acontinuous time process of particles in the lattice obeying the following dynamics:• particles are positioned at vertices of Z≥0 , possibly sharing their location;• if at time t, gi particle are sitting at location i, with rate xi (1 − q gi ) one of them
activates and starts flying to its right. During this rightward motion the particle
crosses each location i0 > i with probability xi0 q gi0 otherwise it stops there;• with a fixed rate r a particle is injected at the leftmost location i = 1.
What we have just informally described is a model for a one dimensional fluid withviscosity (governed by the parameter q) and microscopic inhomogeneities (produced by thexi ). Result of theorem 1.2.1 establishes that, under certain conditions on the parameters
xi , the height function of this process, which is closely related to its cumulative current, inthe stationary state follows typical t1/3 KPZ asymptotic fluctuations distributed according
to the Baik-Rains law. Conditions on the inhomogeneities do not forbid the presence of“slower regions”, where the particles, in the long time tend to accumulate with higher
density.
In mathematical and physical literature “slow bond” problems are of great interest
since it is known that for one dimensional transport, the presence of even minimal microscopic defects has the effect of drastically changing hydrodynamic properties of thesystem [BSS14; BSS17]. In this sense we show that defects of the form described above(which are qualitatively different from those in [BSS14]) do not affect KPZ asymptotic
limit of the process; see also [BP18b].
In recent years KPZ exponents and sometimes fluctuations have been observed directly in a number of remarkable experiments, involving liquid crystals [TS10], collidal
suspensions [Yun+13], bacteria growth [Hue+10] and many more. In particular it became
possible only very recently to directly detect KPZ universality and Baik-Rains universal
27law from systems in steady states [IFT20]. It would be very interesting to see whether
a direct observation of KPZ exponents can be made also in more anthropological context. The model of viscous fluid we introduced above can be thought as a traffic model(for pedestrians more realistically) in an environment where overtaking is allowed anda few roadblocks are present. Steady states in such systems are reached at peak hours
and predictions of theorem 1.2.1 imply that a certain linear combination of the initialstate and the cumulative current through a location will exhibit t1/3 -sized fluctuations,when the time t is sufficiently large. The discussion in section 4.2.1 can then be used toexpress scaling parameters κ, η, γ appearing in theorem 1.2.1 in dependence only of physical observables of the macroscopic system; see also [Spo12]. We leave such interestingdevelopments for future works.
Baik-Rains distribution in diffusion processes
Other new physical implications of our results can be argued considering the Beta polymer
model; see definition 5.2.4. It was observed in [BC16a] that such model of polymers is
equivalent to a simple model of diffusion: the Beta random walk in random environment(RWRE).
A particle X(t) moves on the lattice Z jumping at each time step to its left (resp.
right) neighboring site with probabilities p (resp. 1−p), that are themselves i.i.d. randomvariables (the environment) with Beta distribution. It is clear that in the long time limit,
the√ particle obeys the central limit theorem and the fluctuation of its endpoint are of sizet and converge to the gaussian law. KPZ statistics enter the picture when conditioning
the walker to have an anomalous speed. In this case the large deviations of the random
walk endpoint are corrected at the second order by a random term which scales as t1/3 andconverges to the GUE Tracy-Widom distribution. More precisely [BC16a] proves that
log P{X(t) ≥ tx0 } + I(x0 )t≤ y = FGU E (y),(1.37)lim P
t→∞
C(x0 )t1/3
where P is the probability of the random walk, P is that of the random environment andI(x0 ), C(x0 ) are constants.
A question raised by this result is how other KPZ universal statistics can describe
properties of RWRE and our analysis of the stationary higher spin vertex model (which isa generalization of the Beta polymer and hence of the Beta RWRE) suggests the answer
for the case of the Baik-Rains distribution. This is done by replacing the logarithm of theprobability P with an appropriate expectation of statistics of walks with free endpoint;
see also [BRS19] for a probabilistic approach to the topic. We will cover such interestingdevelopments in a further work.

28

Chapter 2
Spin Hall-Littlewood and spin
q-Whittaker functions
This chapter presents results contained in sections 3,4,5,8 of[BMP19] A. Bufetov, M. Mucciconi, and L. Petrov. “Yang-Baxter random fields and stochastic vertex models”. In: arXiv preprint (2019). arXiv:1905.06815 [math.PR]. To
appear in Adv. Math.
and of sections 2,3 of[MP20] M. Mucciconi and L. Petrov. “Spin q-Whitaker polynomials and deformed quantumToda”. In: arXiv preprint (2020). arXiv:2003.14260 [math.PR]

2.1

Basic notions

2.1.1

Partitions and Signatures

A partition is a sequence of nonnegative integers λ = (λ1 ≥ λ2 ≥ . . .) such that λi = 0
eventually. The last positive entry of the sequence is denoted with λ`(λ) and `(λ) is called
the length of the partition λ while elements λ1 , λ2 , . . . are its parts. Denote by Y the setof all partitions including the empty one λ = ∅ (by agreement, `(∅) = 0). A partition λis naturally represented by its Young diagram, obtained drawing on the plane N × N left
justified sequences of λi boxes; the order we use is matrix-style with i growing downward.
The use of a multiplicative notation is also common:
λ = 1m1 (λ) 2m2 (λ) . . . ,wheremi (λ) = #{j : λj = i}.

For a given λ its transposed partition λ0 is given as
λ0i := #{j : λj ≥ i},i = 1, 2 . . . .

We easily see that the Young diagrams of λ and λ0 are related by reflection with respect
to the main diagonal.
29

Given two partitions µ and λ we say that they interlace if
λ1 ≥ µ1 ≥ λ2 ≥ µ2 ≥ · · · ≥ λk ≥ µk ≥ λk+1 ≥ · · · .

(2.1)

We will use notation µ ≺ λ for interlacing. When λ and µ are such that µ0 ≺ λ0 , we saythat they are transposed interlacing, and use the notation µ ≺0 λ.
A nonnegative signature is a partition with a fixed finite number of parts N . We
will drop the word “nonnegative”, and refer to them simply as “signatures”. Their set isdenoted by
SignN := {λ = (λ1 ≥ · · · ≥ λN ≥ 0) : λi ∈ Z≥0 },N ∈ Z≥0and by agreement, Sign0 = {∅}. To define the transposition in the context of signatures,
introduce the set of boxed signatures
Sign≤NM := {λ = (λ1 ≥ · · · ≥ λM ≥ 0) : 0 ≤ λi ≤ N } ⊂ SignM
S
≤N
≤M
0
and Sign≤N = M ≥0 Sign≤N
M When λ ∈ SignM we have λ ∈ SignN . In this case, theYoung diagram of λ fits inside the the box
Box(N, M ) = {1, . . . , N } × {1, . . . , M },while that of λ0 fits inside Box(M, N ); see Figure 2.1 for an illustration. For pairs ofsignatures λ, µ the notion of interlacing is defined dropping unnecessary inequalities in(2.1).
N

M

Figure 2.1: An example of a signature λ = (5, 4, 1, 1) ∈ Sign≤7
4 . Its transposed signature≤40
is λ = (4, 2, 2, 2, 1, 0, 0) ∈ Sign7 .

2.1.2

Young diagrams as arrow configurations

Functions we introduce in this Chapter are labeled by (skew) Young diagrams which we
represent as configurations of vertical arrows on Z≥0 . Let λ be written in the multiplicative
notation as λ = 1l1 2l2 . . ., where li is the number of parts of λ which are equal to i. By
definition, the arrow configuration corresponding to λ, denoted by |λi, contains li verticalarrows at location i. The number of vertical arrows at 0 is assumed infinite which reflects
the fact that one can append Young diagrams by zeros without changing them. See
Figure 2.2, left, for an illustration. At times it will be convenient to work with the arrow30

configuration corresponding to the transposed of a specific Young diagram λ. In this case
we observe that the number of arrow at position i in |λ0 i is given by λi − λi+1 and we callthis the dual arrow representation of λ.
We will use arrow configurations to construct our special functions following a simple
recipe. Decorate each vertex x of the lattice Z≥0 with a function wx (u) that assigns aweight to arrows configurations around x. Assuming that the weights w are non-zero
only at up-right directed arrow configurations and that they are 1 if no arrow crosses thevertex, we consider then the transfer matrix T(u) = w0 (u)w1 (u)w2 (u) · · · and its matrix
elementsFλ/µ (u) = hµ|T(u)|λi.
Then the functions F are well defined and they admit multi-variable extensions stacking
together multiple rows, as in Figure 2.4, left panel. This procedure guarantees that theyalso follow branching rules (e.g. (2.4) below). To impose symmetries we will requestthat the weights w will be solutions of Yang-Baxter equations, for example we easily seethat the commutation of transfer matrices T(u1 ),T(u2 ) implies that the F’s are symmetricwith respect to the permutation of variables. In the following sections we will need alsofamilies of dual functions G, that we will generally produce using down-right directed
path ensembles; see Figure 2.4, right panel.
∞

0

∞

|λi
1

2

3

4

5

|λi

|µi

Figure 2.2: Left: Configuration |λi of vertical arrows corresponding to the Young diagram
λ = (4, 4, 2, 1, 1, 1). Right: Interlacing of λ with µ = (4, 2, 2, 1, 1, 1).

2.2

Spin Hall-Littlewood functions

2.2.1

Stable spin Hall-Littlewood functions

The first collection of vertex weights we work with is given in Figure 2.3. Along with q,
these weights depend on two quantities u, s ∈ C, which are called the spectral and the spinparameters, respectively. The weights wu,s satisfy the Yang-Baxter equation, see (B.1).
For vertices at the left boundary we set




∞
∞
wu,s
= wu,s (∞, ∅; ∞, 1) = u,wu,s= wu,s (∞, ∅; ∞, 0) = 1, (2.2)∞∞
Both in Figure 2.3 and in (2.2), we attribute weight zero to all configurations which are
not listed. In particular, the following arrow conservation property holds:
wu,s (i1 , j1 ; i2 , j2 ) = 0

unless i1 + j1 = i2 + j2 .
31

(2.3)

g

g

g

g+1

i1

g

g+1

g

g

wu,s (i1 , j1 ; i2 , j2 )1−suq g1−su

u(1−s2 q g )1−su

u−sq g1−su

1−q g+1
1−su

i2
j1

j2

Figure 2.3: In the top row we see all acceptable configurations of arrows entering andexiting a vertex; below we reported the corresponding vertex weights wu,s (i1 , j1 ; i2 , j2 ).
Definition 2.2.1. Consider a sequence of spin parameters s = (s1 , s2 , · · · ) and a spectral
parameter u 6= 1/sx for all x. For µ, λ ∈ Y with µ ≺ λ, the stable spin Hall-Littlewood
function in one variable, denoted by Fλ/µ (u; s), is defined as the total weight (= product ofindividual vertex weights) of the unique configuration of arrows between |µi and |λi as in
Figure 2.2, right panel. Here the individual vertex weights are the wu,sx ’s from Figure 2.3,
and the left boundary weights are (2.2). If µ 6≺ λ, we set Fλ/µ (u; s) = 0. In case all spinparameters are equal sx = s we will drop the explicit dependence on s and write Fλ/µ (u).
In the sequel we will mostly omit the word “stable” (cf. [BMP19, Section 3.3] onconnections to the non-stable functions which were originally defined in [Bor17]), and will
also abbreviate the name to simply the sHL functions.
Define the functions with multiple variables inductively via the branching rule:X
Fλ/ν (uk ; s) Fν/µ (u1 , . . . , uk−1 ; s).
(2.4)
Fλ/µ (u1 , . . . , uk ; s) =νThat is, Fλ/µ (u1 , . . . , uk ; s) is a partition function of ensembles of up-right paths as in
Figure 2.4, left, with height k, spectral parameters u1 , . . . , uk corresponding to horizontal
slices, and boundary conditions |µi, 0∞ , |λi and empty at the bottom, left, up, andright, respectively. The fact that the paths are directed up-right corresponds to the arrowconservation property (2.3). Note that Fλ/µ (u1 , . . . , uk ; s) vanishes unless 0 ≤ `(λ)−`(µ) ≤k, but this condition is not sufficient.
The Yang-Baxter equation implies that Fλ/µ (u1 , . . . , uk ) is symmetric with respect topermutations of the ui ’s, see, e.g., [Bor17, Theorem 3.6]. These functions also satisfy thestability property
Fλ/µ (u1 , . . . , uk , 0; s) = Fλ/µ (u1 , . . . , uk ; s).
(2.5)
For µ = ∅, the stable spin Hall-Littlewood functions admit an explicit symmetrization
formula [BW17; BP18a] which we recall below. When all sx = 0, the stable spin HallLittlewood functions become the usual Hall-Littlewood symmetric polynomials [Mac95,
Chapter III].
Theorem 2.2.2. For any Young diagram λ such that n ≥ `(λ), we have Y `(λ)
n λi 
ui − quj Y Y
ui − sj Y ui
(1 − q)n Xσ.
Fλ (u1 , . . . , un ; s) =(q; q)n−`(λ) σ∈S
u
−
u
1
−
s
u
u
−
s
i
j
j
i
i
λ
i
i=1 j=1i=1
1≤i<j≤n
n

(2.6)

32

Here the symmetric group acts on the indices of the variables ui , but not on λi .
Proof. This theorem is an easy corollary of [BP18a, Thm. 4.14], where a similar formula isstated for the non-stable spin Hall-Littlewood functions. To obtain (2.78) from equation(4.23) of [BP18a] all it takes is to set s0 =0. The denominator (q; q)n−`(λ) in the prefactor
is a consequence of our choice of boundary weights (2.2).
∞

∞

|λi

u6
u5
u4
u3
u2
u1

|κi

v1
v2
v3
v4
v5
v6
1

2 ...

|µi

n
1

2 ...

|νi

n

Figure 2.4: Examples of configurations of up-right and down-right paths used in thedefinitions of Fλ/µ and F∗ν/κ , respectively.
We also report an additional remarkable property of the sHL functions usually called
spatial biorthogonality. The following theorem was proven in [BP18a].
Proposition 2.2.3 (Corollary 7.5 [BP18a]). For any Young diagrams λ, µ, we haveII
Y zi − zjcs (λ)(q; q)n−`(λ)
dz1
dzn
·
·
·
Fλ (z1 , . . . , zn ; s)Fµ (1/z1 , . . . , 1/zn ; s) = 1λ=µ ,(1 − q)n n!
γ 2πiz1γ 2πizn 1≤i6=j≤n zi − qzj(2.7)
where γ is a positively oriented contour encircling 0, q sx for all k ≥ 0, x ≥ 0, and thecontour qγ (its image under the multiplication by q), but not the points s−1
x . Finally thecoefficient cs (λ) is
Y (s2 ; q)`kkcs (λ) =,(2.8)(q; q)`kk≥1
kassuming λ = 1`1 2`2 · · · .

We remark that for the statement of Proposition (2.2.3) we are tacitly assuming thatthe spin parameters allow the construction of the contour γ. Such assumption will always
be satisfied for practical uses of the orthogonality throughout the thesis.

2.2.2

Dual stable spin Hall-Littlewood functions

Let us introduce the dual weights to wu,s from Figure 2.3 as follows:∗
wv,s(i1 , j1 ; i2 , j2 ) =(s2 ; q)i1 (q; q)i2
wv,s (i2 , j1 ; i1 , j2 ).
(q; q)i1 (s2 ; q)i2
33

(2.9)

∗
(i1 , j1 ; i2 , j2 ) vanishes unless i2 + j1 =The arrow conservation law (2.3) implies that wv,si1 + j2 , and as a result the corresponding vertex model produces configurations of directed∗
down-right paths (see Figure 2.4, right). The explicit form of the weights wv,s
is given in∗
Figure 2.5. The weights wv,sat the left boundary are given by the same formulas as in
g

g

g

g+1

i1

g

g+1

g

g

∗ (i , j ; i , j )wv,s1 1 2 2

1−svq g1−sv

1−s2 q g
1−svv−sq g1−sv(1−q g+1 )v
1−sv

i2
j1

j2

Figure 2.5: In the top row we see all acceptable configurations of paths entering and∗(i1 , j1 ; i2 , j2 ).
exiting a vertex; below we reported the corresponding vertex weights wv,s(2.2).
∗
can be obtained from wu,s by substituting u with 1/v, swapping the
The weights wv,svalues of both horizontal edge indices j1 and j2 (that is if j1 = 0, then we change its value
1 and vice versa, and the same for j2 ), and multiplying the result by (v − s)/(1 − vs). Thisswapping construction of the dual weights was instrumental in deriving Cauchy identities
for the sHL functions from the Yang-Baxter equation [Bor17] (a bijectivization of thisargument appeared in [BP19]). In the present thesis we employ a more direct approachwith down-right paths which is better suited for the generalization to spin q-Whittaker∗
functions. The Yang-Baxter equation connecting wu,s and wv,sis recorded in (B.2).
Definition 2.2.4. Consider a sequence of spin parameters s = (s1 , s2 , · · · ) and a spectral
parameter v 6= 1/sx for all x. Fix κ, ν ∈ Y with κ ≺ ν and place the arrow configuration|νi under |κi. Then there exists a unique configuration of horizontal arrows between |κi
and |νi. By definition, a dual stable sHL function in one variable, denoted by F∗ν/κ (v; s),is the total weight of this horizontal arrow configuration, where the individual vertex∗’s from Figure 2.5, and the left boundary weights are the same as inweights are the wv,sx(2.2). If κ 6≺ ν, we set F∗ν/κ (v; s) = 0. We will mainly work with two instances of dualsHL functions, depending on the choice of s. When we take sx = s for all x we will use
the notation F∗ν/κ (v). When we take s1 = · · · = sN −1 = s and sN = sN +1 = · · · = 0 forsome N we will use the notation F•ν/κ (v).
The multivariable generalization F∗ν/κ (v1 , . . . , vk ; s) is defined via the branching rule
exactly as in (2.4). It is the partition function of ensembles of down-right paths as in
Figure 2.4, right, of height k, spectral parameters v1 , . . . , vk corresponding to horizontal
slices, and boundary conditions |κi, 0∞ , |νi, and empty at the bottom, left, top, andright, respectively.
∗
The relation (2.9) between wv,sand wu,s implies thatcs (λ)
Fλ/µ (u1 , . . . , uk ; s) = F∗λ/µ (u1 , . . . , uk ; s),cs (µ)
34(2.10)

where the factor cs was defined in (2.8).
The symmetry of F∗λ/µ (v1 , . . . , vk ; s)) in the vi0 s follows from the symmetry of Fλ/µ . The
dual sHL function also satisfies the same stability property (2.5) as the non-dual one.

2.2.3

Skew Cauchy identities for sHL/sHL pairIn this section we consider the homogeneous versions of the sHL functions, obtained setting
all spectral parameters sx = s, for some s ∈ C. Generalizations in non-homogeneous
settings are straightforward and are contained in [BP18a].
One of the main consequences of the Yang-Baxter equation (either Proposition B.1.1
or Proposition B.1.2) is the skew Cauchy identity for the sHL functions:
Theorem 2.2.5 ([Bor17], [BP18a], [BW17, Section 7.4]). For any two Young diagrams
λ, µ and generic parameters u, v ∈ C such that (u − s)(v − s) < (1 − su)(1 − sv) , wehaveX
1 − quv X
Fλ/κ (u) F∗µ/κ (v).
(2.11)
F∗ν/λ (v) Fν/µ (u) =1 − uv κν

A more refined proof of Theorem 2.2.5 can be given using bijectivization arguments
described below following the approach of [BP19].

2.3

Spin q-Whittaker polynomials

2.3.1

Spin q-Whittaker polynomials

The spin q-Whittaker polynomials are partition functions of up-right path ensembles andthey are labeled by signatures rather than partitions.
We consider up-right directed paths living in the half-quadrant {(i, j) ∈ Z≥0 ×Z≥1 : j ≥i} and we assign weights to a generic vertex (i, j) as:
Wx,s (k) := xk(−s/x; q)k,(q; q)k

if i = 0;(2.12)
Wx,s (i1 , j1 ; i2 , j2 ) := 1i1 +j1 =i2 +j2 1i1 ≥j2 xj2Wx,s (k) :=(−s/x; q)j2 (−sx; q)i1 −j2 (q; q)i2,
(q; q)j2 (q; q)i1 −j2 (s2 ; q)i2

(q; q)k,(−s/x; q)k

if 0 < i < j;(2.13)
if i = j.
(2.14)

Here in (2.12) and (2.14), k ∈ Z≥0 denotes the number of paths going through the vertex,
and in (2.13) the numbers i1 , j1 , i2 , j2 ∈ Z≥0 denote, respectively, the numbers of entering
vertical, entering horizontal, exiting vertical, and exiting horizontal paths to/from thevertex.
35

Remark 2.3.1. One can readily check that the condition i1 ≥ j2 in (2.13) implies that upright path configurations with nonzero global weight are those associated with sequencesof interlacing signatures λ1 ≺ λ2 ≺ . . .. In particular, the configuration in Figure 2.6, left,
has global weight zero.
j

j

4

4

3

3

2

2

1

0

1

2

3

4

5

1

i

0

1

2

3

4

5

i

Figure 2.6: Left: Example of an up-right path ensemble. All paths must be above themain diagonal. Right: Example of a down-right path ensemble with N = 5 and M = 4.
All paths must be inside the rectangle.
Remark 2.3.2. When s = 0, the bulk and the corner weights (2.13)-(2.14) coincide.
More precisely, we have Wx,s (k) = Wx,s (0, k; k, 0) = (q; q)k .
Definition 2.3.3 (Spin q-Whittaker polynomials). For given interlacing signatures µ ≺ λwith µ ∈ Signk and λ ∈ Signk+1 , the skew spin q-Whittaker polynomial in one variable
is the weight of the unique path configuration between µ0 and λ0 at the k-th slice. It is
given bykY(−s/x; q)λi −µi (−sx; q)µi −λi+1 (q; q)λi −λi+1|λ|−|µ|.
(2.15)
Fλ/µ (x) := x(q; q)λi −µi (q; q)µi −λi+i (s2 ; q)λi −λi+ii=1This is clearly a polynomial in x, even though the right corner weight (2.14) is not polynomial. We will often abbreviate the name “spin q-Whittaker” as sqW.

For µ ∈ Signk and ν ∈ Signk+n , we also define n-variable polynomials in a standard
way via branching:X
Fν/µ (x1 , . . . , xn ) =Fκ/µ (x1 , . . . , xn−1 ) Fν/κ (xn ).
(2.16)
κThe polynomials Fν/µ (x1 , . . . , xn ) are partition functions of up-right path ensembles as in
Figure 2.6, left, in a domain with the bottom and the top boundary conditions determinedby µ ∈ Signk and ν ∈ Signk+n , respectively.
We will use the shorthand notation Fλ (x1 , . . . , xn ) ≡ Fλ/∅ (x1 , . . . , xn ), where λ ∈Signn .

Remark 2.3.4. It is important to notice that the number of variables in a sqW polynomialFν/µ is determined by the signatures ν, µ. If ν ∈ Signn+k and µ ∈ Signk , then we can only
evaluate Fν/µ at n variables.
36

2.3.2Comparison with Borodin–Wheeler’s spin q-Whittaker polynomials

It is important to note that our version of the spin q-Whittaker polynomials is different
from the original definition of Borodin and Wheeler [BW17]. Namely, the one-variableskew polynomials in [BW17] have the form
|λ|−|µ|FBWλ/µ (x) = x

Y (−s/x; q)λi −µi (−sx; q)µi −λi+1 (q; q)λi −λi+1i≥1(q; q)λi −µi (q; q)µi −λi+i (s2 ; q)λi −λi+i,(2.17)

where µ ∈ Signk , λ ∈ Signk+1 , and the product over i extends to i = k + 1 with theagreement that λk+2 = µk+1 = 0. That is, our one-variable functions differ from (2.17) as
FBWλ/µ (x) =(−s/x; q)λk+1
Fλ/µ (x).
(s2 ; q)λk+1(2.18)BW
The n-variable polynomials FBW
ν/µ (x1 , . . . , xn ) are defined from Fλ/µ (x) by branchingas in (2.16). They admit a lattice path construction similarly to Fν/µ , but with the rightcorner weights Wx,s replaced by the bulk weights Wx,s .
The Borodin–Wheeler’s spin q-Whittaker polynomials arise from our Fλ as a particularcase:

Proposition 2.3.5. For all λ ∈ Signn we haveFλ (0, x2 , . . . , xn ) = FBW
λ (x2 , . . . , xn ).

(2.19)

Proof. The up-right paths that start at the left boundary at height 1 must immediately
turn up at the right corner at (1, 1). If there are j such paths, their contribution to theglobal weight is Wx1 ,s (j)Wx1 ,s (j) = xj1 . For x1 = 0, this forces no paths to start at height
1. Next, due to the presence of 1c1 ≥k2 in the bulk weight Wx,s , we see that paths started atheight 2, . . . , n cannot reach the diagonal with the special right corner weights. Therefore,
the partition function for Fλ (x1 , x2 , . . . , xn ) with x1 = 0 involves only left boundary andbulk weights, and is thus the same as the partition function for FBW
λ (x2 , . . . , xn ).
Note that FBW
λ (x2 , . . . , xn ) is well-defined for any λ, and vanishes if `(λ), the number ofnonzero parts in λ, exceeds n−1. If `(λ) ≤ n−1, then we can treat λ as an element of Signn
with λn = 0, and then (2.19) holds. Moreover, one readily sees that both sides of (2.19)
vanish if λn > 0. Therefore, any polynomial FBW
can be obtained from our polynomial
λFλ by specializing one of the variables to zero. (By symmetry, see Section 2.3.3 below, we
can specialize to zero any variable, and not necessarily the first one.)

2.3.3

Properties of the spin q-Whittaker polynomials

The fact that the Borodin–Wheeler’s sqW polynomials are symmetric in their variables
follows from the Yang–Baxter equation which we reproduce in (B.12). By looking at(2.18), it is not immediately clear why our version of the sqW polynomials should also be
symmetric. We prove this next.
37

Proposition 2.3.6. For any µ ∈ Signk , ν ∈ Signn+k the polynomial Fν/µ (x1 , . . . , xn ) issymmetric with respect to permutations of its variables xi .
Proof. We use the Yang–Baxter equations (B.12) (B.25) and employ the standard “crossdragging” / commuting transfer matrices argument, cf. [Bor17, Theorem 3.6]. Usingbranching, it suffices to consider the two-variable case. The two-variable polynomial
Fλ/µ (x, y) is a partition function of up-right paths on two consecutive levels, with parameters x, y at the bottom and at the top, respectively, and boundary conditions determinedby λ, µ.
First we use the new relation (B.25) that, as shown in Figure B.7 (a), implies thatswapping the spectral parameters x ↔ y at the right corners makes a cross appear attheir left. Then we sequentially move the cross to the left while swapping the spectral
parameters using the bulk Yang–Baxter equation (B.12), as shown by Figure B.5 (a). We
proceed till the left boundary of the domain.
At the left boundary, we can swap the last two spectral parameters by noticing that
Wx,s (j) =(s2 ; q)∞Wx,s (∞, l; ∞, j),(−sx; q)∞for any l ∈ Z≥0 .

(2.20)

This means that the left boundary weights W also satisfy the Yang–Baxter equation(B.12), and so we can take the cross out of the lattice. This completes the proof.
Our sqW polynomials also satisfy an index shifting property which is the same as forthe classical homogeneous Macdonald polynomials Pλ (·; q, t) [Mac95, VI(4.17)]:
Proposition 2.3.7. For any signature λ ∈ SignN with λN > 0, we haveFλ (x1 , . . . , xN ) = x1 · · · xN Fλ−1N (x1 , . . . , xN ),λ − 1N = (λ1 − 1, . . . , λN − 1).

Proof. First, note that (2.15) implies that the one-variable skew polynomials satisfy theshifting property as
Fν/µ (x) = x F(ν−1k+1 )/(µ−1k ) (x),(2.21)
for any ν ∈ Signk+1 and µ ∈ Signk with νk+1 > 0 (this also implies µk > 0, sinceµk ≥ νk+1 ). Next, we use the expansionXFλ (x1 , . . . , xN ) =Fλ1 (x1 ) Fλ2 /λ1 (x2 ) . . . FλN −1 /λN −2 (xN −1 ) Fλ/λN −1 (xN )λ1 ≺···≺λN −1 ≺λ(2.22)
coming from iterating the branching rule, and apply the shifting property (2.21) to eachof the terms to get the desired result.

Remark 2.3.8. The polynomials FBW
do not satisfy the index shifting property of Propoλsition 2.3.7, which can be seen from (2.18).
On the other hand, the polynomials FBW
satisfy the stability propertyλBW
FBWλ (x1 , . . . , xN −1 , −s) = Fλ (x1 , . . . , xN −1 ),38whereas the polynomials Fλ do not. More precisely, we haveFλ (x1 , . . . , xN −1 , −s) = (−s)λN Fλe (x1 , . . . , xN −1 ),e = (λ1 ≥ · · · ≥ λN −1 ) and this is easily proven since the branching coefficientwhere λQ −1
(2.15) evaluates as Fλ/µ (−s) = (−s)λN N
i=1 1λi =µi .
In the following proposition we use the coefficientc̄(λ) =N
−1Y

(s2 ; q)λi −λi+1
,
(q;
q)
λ
−λ
i
i+1
i=1

λ ∈ SignN .

Proposition 2.3.9. Let |sxi | < 1 for i = 1, . . . , N . Then we have−1X((−s)N x1 . . . xN ; q)∞ (s2 ; q)N∞.
c̄(λ) (−s)|λ| Fλ (x1 , . . . , xN ) =(−sx
1 ; q)∞ . . . (−sxN ; q)∞λ∈Sign(2.23)N

λN =0

Proof. We will use the q-Gauss identity (A.8) that implies
n
X

ak

k=0

(b; q)k (a; q)n−k(ab; q)n=,(q; q)k (q; q)n−k(q; q)n(2.24)

Expand the left-hand side of (2.23) as:
X

X

N

N −1

(−sxN )λ1 −λ1(−s/xN ; q)λN −λN −1
1

λ=λN ∈SignN λN −1 ∈SignN −1
λN
N =0

1

×
×

N
−1
Y

1

(q; q)λN −λN −1(−sxN )

N −1
λN
k −λk

1

(−sxN ; q)λN −1 −λN (−s/xN ; q)λN −λN −1

k=2

k−1

k−1

(−sxN ; q)λN −1
N −1(q; q)λN −1
N −1

× (−s)|λk(q; q)λN −1 −λN

N −1 |kkk(q; q)λN −λN −1kk!

FλN −1 (x1 , . . . , xN −1 ).

2
Summing over λN
1 by means of the q-binomial theorem gives us the factor (s ; q)∞ /(−sxN ; q)∞ .
N
We then sum sequentially over indices λN
2 , . . . , λN −1 and using (2.24) we are left with

(s2 ; q)∞(−sxN ; q)∞X

λN −1 ∈SignN −1(−sxN ; q)λN −1
N −1(q; q)λN −1c̄(λN −1 )(−s)|λN −1 |

FλN −1 (x1 , . . . , xN −1 ),N −1where c̄(λN −1 ) is the result of applying (2.24). Repeating the same procedure we canreduce the previous expression to(s2 ; q)2∞(−sxN ; q)∞ (−sxN −1 ; q)∞((−s)2 xN −1 xN ; q)λN −2
X
N −2
N −2
×
c̄(λN −2 )(−s)|λ | FλN −2 (x1 , . . . , xN −2 ).
(q; q)λN −2
N −2
λ∈SignN −2

N −2

39

Here the reason for the appearance of the product (−sxN )(−sxN −1 ) in the q-Pochhammer−1
symbol is again (2.24), where we also used that λN
N −1 is not necessarily zero (in contrastNwith the first summation over λN ). Continuing inductively, we exhaust all the summations
down to the bottom one over λ11 , from which we recover the factor ((−sx1 ) · · · (−sxN ); q)∞ /(−sx1 ; q)∞ .
This completes the proof.

2.3.4

Skew Cauchy identities for sqW/sHL pair∗
, Wx,s given in (B.26), (B.26) imply the
The Yang-Baxter equations for the weights wv,sfollowing “dual” skew Cauchy identity for the sHL and sqW functions. We recall that thenotation F• was introduced in 2.2.4

Theorem 2.3.10. Fix M ≥ 1. For N > 0, let µ ∈ Sign≤Mand λ ∈ Sign≤M
N
N +1 . Then, wehaveX
1 + vx X
Fν/µ (x) F•ν 0 /λ0 (v) =Fλ/κ (x) F•µ0 /κ0 (v).
(2.25)
1−sv≤M≤M
κ∈SignNν∈SignN +1

For N = 0, we haveXFν (x) F•ν 0 /λ0 (v) = (1 + vx) Fλ (x).

(2.26)

ν∈Sign≤M
1

Note that all the sums in this proposition are over finite sets of signatures, so there
are no convergence issues.
Proof of Theorem 2.3.10. The proof of (2.25) is similar to that of Proposition 2.3.6 as it
also uses a “cross dragging” argument. The summation in the left-hand side of (2.25) is
the partition function of path configurations across two rows of vertices glued together:• the lower row has weights Wx,s , Wx,s , Wx,s and boundary condition µ at the bottomand ν at the top;∗
• the upper one has weights wv,swith si = 0 for i > N , and boundary condition ν atithe bottom and λ at the top.

Recall that the encoding of arrow configurations by signatures is described in detail in
Section 2.1.2.
The Yang–Baxter equation (B.26) implies that the action of swapping weights at therightmost pair of columns, makes a cross weight appear at their left, as shown in Figure B.5 (b). We then push the cross to the left one vertical step at a time, each time swapping the vertex weights and using the Yang–Baxter equation (B.11) as in Figure B.5 (b).
This procedure sequentially turns the left-hand side of (2.25) into the right-hand side.
At the final step, we push the cross out of the lattice at the leftmost site. Using (2.20)
and the left boundary weight (2.2)∗
wv∗ (j) = (1 − sv) wv,s(∞, l; ∞, j),40

for l = 0, 1,

we obtain the combined contribution of the cross vertex weights Rx,v,s (defined in Figure B.4 in the Appendix) corresponding to the two cross configurations and . Theirsum gives the factor (1 + vx)/(1 − sv) in the right-hand side of (2.25), as desired.
The second identity (2.26) can be verified by simply using definition of functions.
Combining the skew Cauchy identities of Theorem 2.3.10, we come to the following
corollary for several variables:
Corollary 2.3.11. For any positive integers N, M, m we haveXFλ (x1 , . . . , xN ) F•λ0 (v1 , . . . , vm ) =

m 
Y
j=1

λ∈Sign≤NM

1
1 − svj

N −1 YN Y
m

(1 + vj xi ).

(2.27)

i=1 j=1

Proof. We use the branching expansion of functions Fλ , F•λ0 and then apply the singlevariable skew Cauchy identities (2.25) and (2.26).

2.3.5

Dual spin q-Whittaker polynomials

Let us also define the dual versions of the sqW weights. These dual weights correspondto down-right lattice paths, and are given by (we use the notation (2.12)–(2.13)):∗,(j) := Wy,s (j);Wy,s(s2 ; q)i1 (q; q)i2
Wy,s (i2 , j1 ; i1 , j2 ).
(q; q)i1 (s2 ; q)i2
W ∗, (i1 , j1 ; i2 ) := 1i2 +j1 =i1∗Wy,s(i1 , j1 ; i2 , j2 ) :=(2.28)(2.29)(2.30)

This choice of vertex weights implies that nonzero global weights are assigned toconfigurations of down-right paths in the grid {0, . . . , N } × {0, . . . , k} which are encodedby sequences of interlacing signatures λ1 ≺ · · · ≺ λk . (Compare this with the transposedinterlacing property for the sHL functions.)
Definition 2.3.12. For given interlacing signatures λ, µ ∈ SignN , the skew dual spin qWhittaker polynomial in one variable F∗λ/µ (y) is the weight of the unique down-right pathconfiguration between µ0 and λ0 at a single row of vertices, with the weights (2.29), (2.28)
and (2.30). Recall that the encoding of arrow configurations by signatures is described in
Section 2.1.2.
An explicit expression for the skew dual sqW polynomial isN−1
Y(−s/y; q)λi −µi (−sy; q)µi −λi+1 (q; q)µi −µi+1∗|λ|−|µ| (−s/y; q)λN −µN
Fλ/µ (y) := y.
(q; q)λN −µN i=1(q; q)λi −µi (q; q)µi −λi+i (s2 ; q)µi −µi+i(2.31)

Observe that F∗λ/µ (y) is a polynomial in y.
Multi-variable extensions F∗λ/µ (y1 , . . . , yk ), where λ, µ ∈ SignN are arbitrary, are defined via branching in the same say as in (2.16). The non-skew functions are F∗ν ≡ F∗ν/0N ,where ν ∈ SignN , and 0N ∈ SignN is the signature with all parts equal to zero.
41

Remark 2.3.13. Like the dual sHL functions and unlike the usual sqW polynomials(cf. Remark 2.3.4), the dual sqW polynomials F∗λ/µ (y1 , . . . , yk ) make sense for any numberof variables k, regardless of the signatures λ, µ.
Proposition 2.3.14. The polynomials F∗λ/µ (y1 , . . . , yk ) are symmetric.
Proof. This follows from the Yang–Baxter equation (B.12) and the sum-to-one property of
the R-matrix (B.14). It suffices to consider swapping two variables. We apply the usual“cross-dragging” argument to exchange spectral parameters of two consecutive rows ofvertices. Similarly to the proof of Proposition 2.3.6, identity (B.12) suffices to swap
spectral parameters from the leftmost column up until the rightmost one. Since the rightboundary weights W ∗, differ from the bulk weights W ∗ , we have to prove that we candrag the cross one more step to the right. We have using the definition of W ∗, that thepartition function near the right wall with the cross vertex is equal toX
X
Rx,y (i1 , i2 ; k1 , k2 )W ∗, (k3 , k2 ; j3 )W ∗, (i3 , k1 ; k3 ) =Rx,y (i1 , i2 ; i1 +i2 −j3 −k, k−j3 ).
k1 ,k2 ,k3k(We used the arrow preservation property i1 + i2 + j3 = i3 .) The right-hand side is equal
to one. Indeed, this sum-to-one property readily follows from identity (2.24). On theotherwithout the cross vertex, the partition function near the right wall is equal to
P hand,
∗,
∗,
W
(i
,
i
(k, i1 ; j3 ). This is also equal to 1, because only the summand with
3 2 ; k)Wk
k = i1 + j3 is nonzero. This completes the proof.

2.3.6

Skew Cauchy identities for sqW/sqW pair

The spin q-Whittaker polynomials also satisfy the following skew Cauchy identity which
follows from the Yang-Baxter equations (B.12), (B.27):
Theorem 2.3.15. For N > 0, let µ ∈ SignN and λ ∈ SignN +1 . Then, for |xy| < 1, wehaveX(−sx; q)∞ (−sy; q)∞ X
Fν/µ (x) F∗ν/λ (y) =Fλ/κ (x) F∗µ/κ (y).
(2.32)
2(s ; q)∞ (xy; q)∞ κ∈Sign
ν∈SignN +1

N

For N = 0, we haveXFν (x) F∗ν/λ (y) =ν∈SignN +1(−sx; q)∞Fλ (x).
(xy; q)∞(2.33)

Proof. For N > 0 this is proven using the same method explained in Theorem 2.3.10 withthe help of identity (B.10) (with q J → −x/s, q I → −y/s) when extracting the cross vertex
weight from the rightmost column. For N = 0 the statement is simply the q-binomial
theorem.
Corollary 2.3.16. Let |xi yj | < 1 for all i = 1, . . . , N , j = 1, . . . , k. Then, we haveN −1 Y
k 
N Y
k
X
Y
(−syj ; q)∞(−sxi ; q)∞∗Fλ (x1 , . . . , xN ) Fλ (y1 , . . . , yk ) =. (2.34)2(s ; q)∞(xi yj ; q)∞j=1
i=1 j=1
λ∈SignN

42

2.3.7

Pieri rules

Pieri type rules for the Borodin–Wheeler spin q-Whittaker polynomials FBW
are given in
λ[BW17]. These are analogs of the classical Pieri type rules for Macdonald polynomials.
The Pieri type rules follow from skew Cauchy identities, and here we present these rules
for our version of the spin q-Whittaker polynomials.
Proposition 2.3.17. Let |xi y| < 1 for all i = 1, . . . , N . Then we have!
X
X (−s/y; q)i
F(i) (x1 , . . . , xN ) Fµ (x1 , . . . , xN ).
Fλ (x1 , . . . xN ) F∗λ/µ (y) =
yi
(q;
q)
i
i≥0
λ∈SignN

Proof. By the skew Cauchy identities of Theorem 2.3.15, we can writeXFλ (x1 , . . . xN ) F∗λ/µ (y) =λ∈SignN(−sy; q)∞(s2 ; q)∞(−sxi ; q)∞Fµ (x1 , . . . , xN ), (2.35)(xi y; q)∞i=1(−sy;q)∞ N −1 QN (−sxi ;q)∞i=1 (xi y;q)∞(s2 ;q)∞and the claim follows by expanding
Proposition 2.3.18. We haveXN −1 YNNX

Fλ (x1 , . . . xN ) F•λ0 /µ0 (v) =!

F•(i) (v) F1i (x1 , . . . xN ) Fµ (x1 , . . . , xN )i=0λusing (2.34).

Proof. By the skew Cauchy identities of Theorem 2.3.10, we haveXFλ (x1 , . . . xN ) F•λ/µ (v) =λ∈SignN

and the claim follows by expanding1
1 − sv

N −1 YN
(1 + xi v) Fµ (x1 , . . . , xN ),(2.36)i=i

N −1 QN
1
i=i (1 + xi v) using (2.27).
1−sv

Pieri type rules of Propositions 2.3.17 and 2.3.18 are eigenrelations on the spin qWhittaker polynomials in the label variable. Indeed, define operators HsqW , HsHL as
X
Xf (λ) F•λ0 /µ0 (v).
(2.37)(HsqW f )(µ) =f (λ) F∗λ/µ (y),(HsHL f )(µ) =λλThen these operators act diagonally on spin q-Whittaker functions f (λ) = Fλ (x1 , . . . , xN ),with respective eigenvalues
X
i≥0

yi

(−s/y; q)i
F(i) (x1 , . . . , xN )(q; q)i

and

N
X
i=0

43

F•(i) (v) F1i (x1 , . . . xN ).

2.4

Analytic continuation

2.4.1

A common generalization of skew Cauchy identities

The skew Cauchy identities given in the previous sections admit a (partial) common
generalization which can be viewed as an analytic continuation. In [Bor17], [BP18a],
principal specializations of non-stable spin Hall-Littlewood functions were considered.
They are obtained by setting spectral parameters to finite geometric progressions of ratio
q. In our context, define(J ,...Jn )(u1 , . . . , un ) = Fλ/µ (u1 , qu1 , . . . , q J1 −1 u1 , . . . , un , qun , . . . , q Jn −1 un ; s)(2.38)(I ,...Im )(v1 , . . . , vm ) = F∗λ/µ (v1 , qv1 , . . . , q I1 −1 v1 , . . . , vm , qvm , . . . , q Im −1 vm ; s).

(2.39)

1
Fλ/µ
1
Gλ/µ

It is a consequence of the fusion procedure (dating back to [KRS81] and briefly recalled(J1 ,...Jn )in Appendix B.1.2), that we can view Fλ/µ(u1 , . . . , un ) as a partition function in a“smaller” vertex model obtained by attaching together n (instead of J1 + . . . + Jn ) rows
(J )(I1 ,...Im )with fused weights wukk,s , (B.3) where k = 1, . . . , n. Analogously Gλ/µ(v1 , . . . , vm ) are∗,(I )

partition functions of a vertex model with fused weights wvk ,sk (B.6). As usual, at theleftmost column of these lattices we place infinitely many vertical paths.
As a result of the vertex model constructions and of the fact that the weights w(J) ,w∗,(I) degenerate both to w, w∗ and W, W ∗ (see Appendix B.1.3), we see that (2.38), (2.39)
interpolate between the spin Hall-Littlewood functions and the Borodin-Wheeler’s spin
q-Whittaker functions. These functions (2.38)–(2.39) satisfy the following general skew
Cauchy identity which we state for an appropriate “analytic” range of parameters:
Theorem 2.4.1. Fix m, n ∈ Z≥0 . Take q ∈ (0, 1), and let si 6= 0, ui , q Ji , vj , q Ij be complex
parameters satisfying|sj |, |uk |, |vl |, |q Jk uk |, |q Il vl |,q i v l − sj
q i uk − sj,<δ
1 − q i sj uk
1 − q i svl

for all k, l, i, j,(2.40)

for sufficiently small δ > 0 which might depend on m, n, but not on the other parametersof the functions.1 Then we have
X (J ,...J )(I1 ,...Im )n1Fν/µ(u1 , . . . , un )Gν/λ(v1 , . . . , vm )
ν

=

n Y
m
Y
(uk vl q Il ; q)∞ (uk vl q Jk ; q)∞ X(uk vl ; q)∞ (uk vl q Il +Jk ; q)∞k=1 l=1(J ,...,Jn )1Fλ/κ(I ,...,Im )1(u1 , . . . , un )Gµ/κ

(v1 , . . . , vm ).

κ(2.41)

Remark 2.4.2. This identity immediately degenerates to the skew Cauchy identities forspin Hall-Littlewood and Borodin-Wheeler’s spin q-Whittaker polynomials after appropriately specializing the parameters uk , vl and q Jk , q Il .
1

Here and below in this section we treat q Jk and q Il as separate symbols independent of q. At thesame time, for Jk a positive integer, q Jk means the corresponding power of q, and same for q Il .

44

Remark 2.4.3. It is not true that the principal specializations of the spin Hall-Littlewood
functions degenerate to our version of the spin q-Whittaker polynomials and this is because
it does not seem possible to construct the corner weights W via fusion. Nevertheless, sincemost of our arguments and construction work the same for both cases, in order to keep
the notation compact, we will sometime refer to the sqW polynomials as a specialization(J1 ,...,Jn )of the functions Fλ/µ.
The proof of Theorem 2.4.1 requires an absolute convergence result for spin HallLittlewood functions with principal specializations:
Proposition 2.4.4. Fix n ∈ Z≥1 . Let q ∈ (0, 1). Take s 6= 0, ui , q Ji to be complex
parameters satisfying|sj |, |uk |, |q Jk uk |,q i u k − sj<δ
1 − q i sj uk

for all k, i,(2.42)

for some δ = δn > 0 (which might depend on n). Then for all Young diagrams µ we haveX
(J1 ,...Jn )(u1 , . . . , un ) < ∞.
(2.43)
Fλ/µ
λ : µ⊆λThe proof of Proposition 2.4.4 will be given later in Section 2.4.2. First we use its
result to justify the general Cauchy identity (2.41):
Proof of Theorem 2.4.1 modulo Proposition 2.4.4. By Theorem 2.2.5, identity (2.41) holds(J1 ,...,Jn )(I1 ,...,Im )in case J1 , . . . , Jn , I1 , . . . , Im are positive integers. Functions Fλ/µ, Gλ/µare finite(J )∗,(I )

sums of finite products of weights wukk,s , wvl ,s l which are rational functions of q Jk , q Il .
(J1 ,...,Jn )(I1 ,...,Im )Therefore, Fλ/µ, Gλ/µadmit an extension to generic complex numbers q Jk , q Il .
This implies that the right-hand side of (2.41) extends to q Jk , q Il in a complex region, too,since the sum over κ is finite (it ranges over κ ⊆ µ, λ).
The summation in the left-hand side of (2.41) has infinitely many terms as the only
condition on ν is that µ, λ ⊆ ν. Therefore, to show that it can be extended to parameters
q Jk , q Il in a complex region we need a result of absolute convergence of the sum overν. Under assumptions (2.40), this is a consequence of Proposition 2.4.4. Therefore, theleft-hand side of (2.41) can be extended, too.
The equality between the two sides of (2.41) in a wider region (2.40) follows because
these expressions agree for infinitely many values of Jk , Il , namely, positive integers: if|uk | < δ, then |uk q Jk | < δ for all Jk ∈ Z≥1 . This completes the proof.
Despite the fact that the general skew Cauchy identity (Theorem 2.4.1) offers a unified
description of all skew Cauchy structures we study, throughout the text we still consider
possible degenerations separately. There are several reasons for this. First, the spin
Hall-Littlewood and the spin q-Whittaker functions are more basic from an algebraic
standpoint (see, e.g., Section 2.6 where we describe difference operators diagonalized bythese functions). Second, when u, q J , v, q I are general parameters, it is difficult to give aprobabilistic interpretation of the random fields — the positivity of the measure obtainedby multiplying F and G is in general not guaranteed.
45

2.4.2Absolute summability

We now turn to the proof of the absolute summability result of Proposition 2.4.4. This
proof requires explicit expressions for the fused weights which can be found in Chapter B.
We begin with a number of preliminary estimates, and assume that q ∈ (0, 1) throughout
the subsection.
(J)

Lemma 2.4.5. Consider the fused weights wu,s (i1 , j1 ; i2 , j2 ) defined in (B.3), with γ =q J ∈ C and i1 , j1 , i2 , j2 ∈ Z≥0 . Let s 6= 0 and δ = max{|s|, |u|, |γu|} < 1. Then(J)
(i1 , j1 ; i2 , j2 ) ≤ (min{i1 , j2 } + 1) Cδ j2 ,wu,s(2.44)

where C is a positive constant independent of the vertex configuration (i1 , j1 ; i2 , j2 ).
Proof. Expand w(J) combining (B.3) and (A.9) as1(−1)i1 +j2 q 2 i1 (i1 −1+2j1 ) sj2 −i1 ui1 (u/s; q)j1 −i2 (q; q)j1
(q; q)i1 (q; q)j2 (su; q)j1 +i1
i1
X
qk
×
(q −i1 ; q)k (q −i2 ; q)k (suγ; q)k (qs/u; q)k(q; q)kk=0(2.45)× (q k s2 ; q)i1 −k (q 1+j2 −i1 +k ; q)i1 −k (γq 1−i2 −j2 +k ; q)i1 −k .

First, the factor(−1)i1 +j2 (q; q)j1 (suγ; q)k (q k s2 ; q)i1 −k(q; q)i1 (q; q)j2 (su; q)j1 +i1 (q; q)kis bounded in absolute value by a constant independent of i1 , j1 , i2 , j2 . The q-Pochhammer
symbol (q 1+j2 −i1 +k ; q)i1 −k vanishes unless k ≥ i1 − j2 and its contribution is bounded inabsolute value by 1. The remaining factors are1

q 2 i1 (i1 −1+2j1 ) sj2 −i1 ui1 (u/s; q)j1 −i2 q k (q −i1 ; q)k (q −i2 ; q)k (qs/u; q)k (γq 1−i2 −j2 +k ; q)i1 −k .
Rewriteq i1 j1 (q −i2 ; q)k (γq 1−i2 −j2 +k ; q)i1 −k =

i1 Y
−k−1

k−1
Y
l=0

(q j1 − q −i1 +j2 +l )l=0

(q j1 − γq −l ),(2.46)

where we used the arrow conservation property, andqi1 (i1 −1)/2+k

(q

−i1

; q)k

i1 Y
−k−1
l=0

(q j1 − γq −l ) = (−1)i1 γ i1 −k (q i1 −k+1 ; q)k (q j1 /γ; q)i1 −k .

(2.47)

The factors (−1)i1 (q i1 −k+1 ; q)k are bounded in the absolute value. By substituting (2.46),(2.47) into (2.45), we see that it remains to address the termsj2 −i1 ui1 (u/s; q)j2 −i1 (qs/u; q)k γ i1 −k (q j1 /γ; q)i1 −k
46

k−1
Y
l=0

(q j1 − q j2 −i1 +l ).

(2.48)

We consider two cases based on the sign of j2 − i1 .
Q
j1
j2 −i1 +lCase j2 ≥ i1 . The factor k−1) in (2.48) is bounded by a constant indel=0 (q − qpendent of the vertex configuration. The remaining terms are
sj2 −i1 (u/s; q)j2 −i1 · ui1 (qs/u; q)k · γ i1 −k (q j1 /γ; q)i1 −k .
Distributing the factors s, u, and γ into the q-Pochhammer symbols, we can bound theabove expression by const · δ j2 , where j2 is the total number of terms in the product. Notethat the estimate works uniformly for small s, u, γ, too.
Case i1 > j2 . Rewriting the q-Pochhammer symbol with the negative index (cf. (A.1))
and using the fact that i1 − j2 ≤ k ≤ i1 , we have
(2.48) = (−1)i1 −j2 uj2 (q i1 −j2 +1 s/u; q)k−i1 +j2 γ i1 −k (q j1 /γ; q)i1 −k q

 k−1
Y(q j1 − q j2 −i1 +l ).

i1 −j2 +1
2

l=0) Qk−1 (q j1 − q j2 −i1 +l ) is bounded. The contribution of the term

i1 −j2 +1
2

The term (−1)i1 −j2 q (

l=0

uj2 (q i1 −j2 +1 s/u; q)k−i1 +j2 γ i1 −k (q j1 /γ; q)i1 −k
is bounded by const · δ j2 similarly to the previous case.

We see that (2.45) can be written as a sum of terms bounded by const · δ j2 . Because
the number of terms is≤ min{i1 , i2 } + 1 − max{0, i1 − j2 } ≤ min{i1 , j2 } + 1,
we get the desired bound.

Lemma 2.4.6. Let

qnu − s< δ < 1.
supnn∈Z≥0 1 − q su

(2.49)(J)|wu,s(0, j1 ; i2 , j2 )| ≤ C(i2 ) δ j2 ,(2.50)

Then
where C(0) = 1 and C = supk {C(k)} < ∞.
Proof. This bound follows from (B.3): after setting i1 = 0, the q-hypergeometric series
disappears, and we use the definition of δ to estimate the prefactor.
For a Young diagram λ, let mi (λ) be the number of parts in λ which are equal to i.
Lemma 2.4.7. Let sj 6= 0, u, q J be complex parameters such that|sj |, |u|, |q J u|,

q i u − sj< δ,
1 − q i sj u
47

for all i, j,(2.51)

∞

|νi

|µi

Figure 2.7: The decomposition of the Young diagram ν as a free superposition of η (black
paths) and κ (blue paths) used in the proof of Lemma 2.4.8.
for some δ ∈ (0, 1). Then there exists C ≥ 1 such that for any two Young diagrams µ, λwe haveY(J)Fλ/µ (u) ≤ C M (λ,µ) (mi (µ) + 1)δ |λ|−|µ| ,(2.52)i≥1where

M (λ, µ) = 1 + #{i : mi (µ) 6= 0 or mi (λ) 6= 0}.

(2.53)

Proof. It suffices to assume that µ ⊆ λ (i.e., µi ≤ λi for all i), otherwise the skew function
vanishes. We have
Y

X
∞
(J)
(J)
(J)
wu,s
(ml (µ), jl−1 ; ml (λ), jl ),(2.54)
wu,s
Fλ/µ (u) =j0
l∞j ,j ,···≥0
l≥1
0

1

where the infinite sum has only one nonzero term due to arrow preservation. From (B.5)
we have for the leftmost vertex


∞
(J)
≤ Cδ j .
(2.55)
wu,sj∞
Lemmas 2.4.5 and 2.4.6 provide estimates for the remaining vertex weights: they are allbounded by Cδ jl , except if both ml (µ) = ml (λ) = 0 when the bound is given by δ jl .
Lemma 2.4.8. Let 0 < δ < 1, C ≥ 1, 1 ≤ A < δ −1 , and M (λ, µ) be as in Lemma 2.4.7.
Then for any Young diagram µ we haveXY`(µ)
C M (ν,µ) δ |ν|−|µ| A`(ν) (mi (ν) + 1) ≤ C1 A1 ,(2.56)
ν : µ⊆νi≥1where C1 , A1 ≥ 1 are constants.
Proof. The sum over ν can be visualized as a sum over path configurations in a rowof vertices as in Figure 2.7. We distinguish the paths coming from the configuration µ(black dashed in Figure 2.7) and when they originate at the leftmost vertex (blue solid in
Figure 2.7). Calling κ the Young diagram generated by the blue paths and η the Youngdiagram generated by the black paths we can write ν = κ ∪ η (this decomposition is not48

unique). The sum in the left-hand side of (2.56) is dominated by a sum where κ and ηvary independently, and therefore we haveXY
M (κ,∅) |κ| `(κ)(mi (κ) + 1)
lhs (2.56) ≤
C
δ A
κ

×

 X

i≥1

C

M (η,µ) |η|−|µ|

δ

`(µ)

A


Y
(mi (η) + 1) .

(2.57)i≥1η : µ⊆η`(η)=`(µ)

We
P estimate separatelyPthe two factors in (2.57), starting with the first one. Since `(κ) =i mi (κ) and |κ| =i≥1 i mi (κ), the summation over κ can be rewritten as follows.
Separate the term κ = ∅. In the remaining sum, first select κ1 ≥ 1 and its multiplicityr ≥ 1; then for each i = 1, . . . , κ1 − 1, select a multiplicity mi ≥ 0. Summing over allthese possibilities, we have
!
κY
1 −1
X X
X
C+
C
δ rκ1 C(r + 1)ArC 1mi >0 (mi + 1)δ imi Ami .
κ1 ≥1

mi ≥0

i=1

r≥1

Simplifying the geometric summations and using the fact that Aδ < 1, we can reduce theabove sum to
1 −1 
X ACδ κ1 (2 − Aδ κ1 ) κY1
1−C 1−
C +C(1 − Aδ κ1 )2(1 − Aδ i )2
i=1
κ ≥1
1

For all i ≥ i0 , where i0 depends on C, A, δ the i-th term in the product is less than δ −1(because δ < 1 and the term goes to 1 as i → ∞). This implies that the above sum isconvergent and thus is estimated from above by a constant.
We now address the second factor in the right-hand side of (2.57). We can again
bound the sum over η by a superposition of `(µ) noninteracting paths starting at µi . Thisimplies that the sum over η in (2.57) is dominated by`(µ)


`(µ)
Y
X
δ
M (ki ,µi )ki −µi2.
C
2Aδ
= 2C A 1 + C
1−δ
i=1 k ≥µ
iiThis completes the proof.
(J ,...J )n
1
Proof of Proposition 2.4.4. We first expand Fλ/µ(u1 , . . . , un ) in (2.43). Utilizing thebranching rule and the triangle inequality, we can estimatelhs (2.43) ≤X

λ1 ,...λn :µ⊆λ1 ⊆···⊆λn

n
Y

(J )

Fλii/λi−1 (ui ) .

(2.58)

i=1

In order to evaluate the previous nested summation we start from the most external term.
For fixed λn−1 we have
X
Y
X
n n−1
(J )
|λn |−|λn−1 |Fλnn/λn−1 (un ) ≤(mi (λn−1 ) + 1)
C M (λ ,λ ) δ1,
λn : λn−1 ⊆λni≥1λn : λn−1 ⊆λn49

where we used bound of Lemma 2.4.7 for some δ1 ∈ (0, 1). We can further estimate the sum
Q`(λn−1 )over λn with the help of Lemma 2.4.8, and obtain the bound i≥1 (mi (λn−1 )+1)C1 A1.
−1
Replacing δ1 by a smaller value 0 < δ2 < A1 if needed and multiplying this bound by(Jn−1 )
the next term Fλn−1/λn−2 (un−1 ) in (2.58), we can apply Lemma 2.4.7 and then sum overn−1λwith the help of Lemma 2.4.8. Iterating this procedure finitely many times, we get
the desired statement with a sufficiently small δn > 0.

2.5

Scaled geometric specializations

In this section we introduce a third specialization — the scaled geometric one — of thegeneral fused functions from the previous section. This specialization allows to include
into our analysis stochastic particle systems with more general initial configurations.

2.5.1

Definition of scaled geometric specializations

We now introduce yet another specialization of (2.38), (2.39) which admits a meaningfulprobabilistic interpretation — it corresponds to two-sided stationary initial conditions forstochastic particle systems on the line arising as marginals of our Yang-Baxter fields.
Definition 2.5.1 ([BP18a]). The scaled geometric specialization with parameter α of thespin Hall-Littlewood function is given by
Fλ/µ (ᾰ; s) := lim Fλ/µ (−α, −αq, . . . , −αq J−1 ; s)→0q J =1/

.

(2.59)c(λ)
The dual analog of F(ᾰ; s) is given by the conjugation F∗λ/µ (ᾰ; s) = c(µ)
Fλ/µ (ᾰ; s) asin (2.10). The skew functions in multiple variables Fλ/µ (ᾰ1 , . . . , ᾰN ; s) are defined in astandard way using the branching rules as in (2.4), and similarly for dual functions.

The functions Fλ/µ (ᾰ; s), F∗λ/µ (β̆; s) also admit a lattice construction with the vertex
weights(J)∗,(I)∗
w̆α,s := lim w−α,s,w̆β,s:= lim w−β,s
.
→0

→0

q J =1/

q I =1/The expressions for these weights are given in Section B.1.4. The functions Fλ/µ (ᾰ; s), F∗λ/µ (β̆; s)vanish unless µ ⊆ λ (which means that µi ≤ λi for all i).
By adding the scaled geometric specialization to our symmetric functions, we candefine mixed specializations Fλ/µ (u1 , . . . , un , ᾰ1 , . . . , ᾰN ). They are obtained through thebranching rules as
XFλ/µ (u1 , . . . , un , ᾰ1 , . . . , ᾰN ; s) =Fλ/κ (ᾰ1 , . . . , ᾰN ; s) Fκ/µ (u1 , . . . , un ; s),κand similarly for the dual functions. By the Yang-Baxter equations (Section B.1.4),
these functions are symmetric in all the N + n variables. We will also sometimes use
50the notation sHL(u) and sg(α) to denote the two types of specializations of the generalsymmetric functions (2.38), (2.39). Similarly we will at times say that wu,s and ᾰα,s are(J)
respectively the sHL(u) and sg(α) specializations of the weight wu,s .
Since the dual spin q-Whittaker functions can be obtained from F∗ from fusion (unlike
the non-dual F), we can define their version with mixed specializations asXF∗λ/µ (y1 , . . . , yn , ᾰ1 , . . . , ᾰN ) =F•λ/κ (ᾰ1 , . . . , ᾰN ) Fκ/µ (y1 , . . . , yn ),(2.60)
κ

where λ ∈ Signk and by agreement F•λ/κ (ᾰ1 , . . . , ᾰN ) is equal to F∗λ/κ (ᾰ1 , . . . , ᾰN ; s), withspin parameters s1 = · · · = sk−1 = s, sk = sk+1 = · · · = 0

2.5.2

Mixed Skew Cauchy identities

The scaled geometric specializations allow to extend the skew Cauchy identities consideredin previous sections. We will list the results without going into the proofs, that in all cases
follow arguments seen in the sections above.
In the following statements we will assume that q ∈ (0, 1), s ∈ (−1, 0].
Proposition 2.5.2. Take u ∈ C such that |s(s − u)| < |1 − su|. Then for all α we have
X
X
Fν/µ (u)F∗ν/λ (ᾰ) = (1 + αu)Fλ/κ (u)F∗µ/κ (ᾰ).
(2.61)
κνProposition 2.5.3. Take α, x such that |αx| < 1. Then, for all µ, λ ∈ SignN , we haveX(−sα; q)∞ X
Fν/µ (x)F•ν/λ (ᾰ) =Fλ/κ (x)F•µ/κ (ᾰ),(2.62)(αx;q)∞ν∈Signκ∈SignN −1

N +1

if N > 1. When N = 1 we haveX
Fν/µ (x)F•ν/λ (ᾰ) =ν∈SignN +1

1
Fλ/κ (x).
(αx; q)∞(2.63)

We finally report the skew Cauchy identitiy for pairs of scaled geometric specializations.
Proposition 2.5.4. Take α, β such that |αβ| < 1. Then, for all µ, λ, we have
X
X
1
Fν/µ (ᾰ)F∗ν/λ (β̆) =Fλ/κ (ᾰ)F∗µ/κ (β̆).
(αβ;
q)
∞
ν
κ

2.6

(2.64)

Difference operators

In this Section we prove that the (stable) spin Hall-Littlewood and the spin q-Whittaker
functions are eigenfunctions of certain (q-)difference operators acting on symmetric functions. In this section we denote the quantization parameter in the sHL functions by tinstead of q because the sHL eigenoperators are the same as in the Macdonald case (recall
that for s = 0, the sHL functions become the usual Hall-Littlewood symmetric polynomials, which are the q = 0 degenerations of the Macdonald symmetric polynomials).
51

2.6.1

Eigenrelations for the spin Hall-Littlewood functions

Consider the space of symmetric rational functions in u1 , . . . , un . Let the operator Tq,ui
on this space beTq,ui f (u1 , . . . , un ) = f (u1 , . . . , ui−1 , qui , ui+1 , . . . un ),(2.65)

that is, it acts by multiplying the variable ui by q. In this subsection we will use the q = 0
version, T0,ui . Note that this operator acts only on rational functions whose denominators
do not contain positive powers of ui .
Definition 2.6.1 (Hall-Littlewood difference operators). For 1 ≤ r ≤ n, let the r-th
Hall-Littlewood difference operator beX  Ytui − uj∗
T0,I ,(2.66)Dr :=ui − uji∈Iwith T0,I =Q

I⊂{1,...,n}|I|=rj∈{1,...,n}\Ii∈I T0,ui .

The Hall-Littlewood operators are the q = 0 cases of the Macdonald difference operators [Mac95, Chapter VI.3] (the latter are obtained by taking Tq,ui in (2.66) insteadof T0,ui ). The operators Dr are diagonal in the Hall-Littlewood symmetric polynomials
Fλ |s=0 :2∗Dr Fλ (u1 , . . . , un )|s=0 = er (1, t, . . . , tn−`(λ)−1 ) Fλ (u1 , . . . , un )|s=0 ,(2.67)
where the eigenvalues are given in terms of er (u1 , . . . un ), the r-th elementary symmetricpolynomial:Xer (z1 , . . . , zN ) =zi1 · · · zir .
(2.68)
1≤i1 <···<ir ≤N

In particular, er (z1 , . . . , zN ) = 0 if r > N .
In the following Theorem we extend (2.67) to the spin Hall-Littlewood symmetricfunctions:
Theorem 2.6.2. For all Young diagrams λ and n ∈ Z≥1 we have∗

Dr Fλ (u1 , . . . , un ; s) = er (1, t, . . . , tn−`(λ)−1 ) Fλ (u1 , . . . , un ; s).

(2.69)

Remark 2.6.3. Certain difference operators acting diagonally on the non-stable spin
Hall-Littlewood symmetric functions were considered in [Dim18].
In order to prove Theorem 2.6.2 we make use of a preliminary lemma.
Lemma 2.6.4. We have YX  Y ui − tuj ui − tuj
∗ X
n−1
Dr
σ
= er (1, . . . , t )
σ
.
u
−
u
u
−
u
i
j
i
j
1≤i<j≤n1≤i<j≤n
σ∈S
σ∈S
n

2

(2.70)n

We have Fλ (u1 , . . . , un )|s=0 = Qλ (u1 , . . . , un ; t) in the standard notation of [Mac95, Chapter III].

52

Proof. This is the λ = ∅ case of the known Hall-Littlewood relation (2.67). Notice thatthe symmetrized sum in fact does not depend on the variables u1 , . . . , un :X  Y ui − tuj (t; t)n
=
,
σ
n
u
−
u
(1
−
t)
i
j
1≤i<j≤nσ∈Snsee [Mac95, Chapter III.1, formula (1.4)].
Proof of Theorem 2.6.2. For a fixed Young diagram λ we defineYui − tuj
,
A=
u
−
u
i
j
1≤i<j≤n

B=


λi 
n Y
Y
u i − sj
i=1 j=1

1 − sj ui

,

C=

`(λ)
Y

ui
.
u
−
s
i
λ
i
i=1

With this notation, using (2.6), the left-hand side of (2.69) can be written as
cλ

X

Y

i∈I
I⊂{1,...,n}j∈{1,...,n}\I|I|=rX
tui − uj
T0,I
σ {ABC} ,ui − ujσ∈S(2.71)nwhere cλ = (1 − t)n /(t; t)n−`(λ) . We first observe that(
σ{C} if I ⊆ {σ`(λ)+1 , . . . , σn },T0,I σ{C} =0otherwise.
Therefore, we can reduce the sum over the symmetric group in (2.71) to permutations σsuch that I ⊆ σ({`(λ) + 1, . . . , n}). Moreover, we see that the claim of Theorem 2.6.2
follows for r > n − `(λ) since both sides of (2.69) vanish. Thus we will now assume thatr ≤ n − `(λ).
For a given permutation σ define the ordered sets Vσ , Wσ asVσ = σ({1, . . . , `(λ)}) = v1 , . . . , v`(λ) ,Wσ = σ({`(λ) + 1, . . . , n}) = {w1 , . . . , wn−`(λ) } = I ∪ K,
and rewrite (2.71) as
cλ

X

X

I⊂{1,...,n} σ∈Sn
I⊆Wσ
|I|=r

σ {BC}Yi∈I
j∈{1,...,n}\I

tui − uj
T0,I σ {A} ,ui − uj(2.72)

where we used the fact that σ{BC} only depends on variables uj for j ∈ Vσ . We now
focus on the remaining factors. For two disjoint or coinciding ordered sets S1 , S2 denote

53

P (S1 , S2 ) :=Yi∈Ij∈{1,...,n}\IQ

ui −tuji∈S1 , j∈S2 ui −uj . When S1 = S2 , the product is only over i < j. We havetui − uj
T0,I σ {A} = P (I c , I)P (Vσ , Vσ )P (Vσ , K)ui − uj× T0,I P (I, K)P (I, I)P (Vσ , I)P (K, I)P (K, K)= P (Vσ , Vσ )P (Vσ , Wσ )P (K, I) T0,I P (I, K)P (I, I)P (K, K)
Y
Y ui − tujYYuvi − tuvj
uwi − tuwj
ui − tuj=
T0,I.
uvi − uvj i∈V , j∈W ui − uj
ui − uj
uwi − uwj
1≤i<j≤`(λ)σ

i∈Wσ \I,j∈I

σ1≤i<j≤n−`(λ)

In the above calculation we used the fact that T0,I acts on P (S, I), S 6= I, by turning
it into one. The action T0,I P (I, I) does not make sense before the symmetrization (i.e.,summation over σ), and so we do not apply T0,I to this expression just yet. In the lastline, the first two products are independent of I and of the ordering of Wσ , and the lasttwo products are independent of the ordering of Vσ . Therefore, we can rearrange the twosummations in (2.72) asX
X Y uv − tuw  Y
uvi − tuvj
BC
τ
cλ
u
u
v − uwvi − uvjv∈Vτ ∈S
V ⊆{1,...,n}|V |=`(λ), W =V c

`(λ)

×

1≤i<j≤`(λ)

w∈W

X

Y tui − uw
XT0,Iui − uw
i∈I
σ∈S

I⊂W
|I|=r w∈W \Iσn−`(λ)

Y

1≤i<j≤n−`(λ)

 (2.73)
uwi − tuwj.
uwi − uwj

The permutations τ, σ permute the variables uvi , uwj , vi ∈ V , wj ∈ W , acting respectively
on indices i and j. We can now employ Lemma 2.6.4 to transform the second line of(2.73) intoXYuwi − tuwjn−`(λ)−1.
(2.74)er (1, t, . . . , t)σuwi − uwjσ∈S
1≤i<j≤n−`(λ)

n−`(λ)

Therefore,XX

uwi − tuwj
uwi − uwjYlhs (2.69) = er (1, t, . . . , tn−`(λ)−1 ) cλ

×σ



Y

1≤i<j≤n−`(λ)

Y uv − tuwτ ∈S`(λ) v∈VV ⊆{1,...,n}|V |=`(λ), W =V c σ∈Sn−`(λ) w∈W τ

1≤i<j≤`(λ)

uv − uwuvi − tuvj
BC .
uvi − uvj

(2.75)The summations in the right-hand side of this last expression (along with the factor cλ )are easily rearranged into the symmetrized sum (2.6) producing the spin Hall Littlewood
function Fλ .

54

2.6.2

Eigenrelations for F•λThe operator we introduce next depends on the number of variables M and on an additional positive integer N . Moreover, this operator acts only on a certain subspace ofrational functions. Namely, let V(M ) be the space of symmetric rational functions in M
variables v1 , . . . , vM of degree ≤ 1 in each variable. That is, its elements are functions
f (v1 , . . . , vM ) = a(v1 , . . . , vM )/b(v1 , . . . , vM ), where a and b are polynomials such thatdegvi (a) − degvi (b) ≤ 1 for all i = 1, . . . , M . One readily sees that V(M ) is a linear space.
The dual sHL functions F•ν (v1 , . . . , vM ) belong to V(M ), see (2.78).
Definition 2.6.5. For positive integers M, N define the dual s-deformed Macdonald operator by
M Y
MXvj − tvl•
Cj,N ,(2.76)
D1,N :=v−v
jl
j=1 l=1
l6=j

where
Cj,N := vjvj − s1 − svj

N −1(−s)N −1 lim ε Tε−1 ,vj .
ε→0The limit limε→0 ε Tε−1 ,vj is well-defined on V(M ), so D•1,N acts in the space V(M ).
Theorem 2.6.6. For any boxed signature λ ⊆ Box(N, M ) (recall that this is Sign≤NM ), wehave
0D•1,N F•λ = e1 (1, t, . . . , tλN −1 ) F•λ ,(2.77)
where λ0 is the transposed signature. In particular, λ0N = # {i : λi = N }.
For the proof it is useful to specialize symmetrization formula (2.6) to the case offunctions F• .
Proposition 2.6.7. Let λ ∈ Sign≤N
M , then for all k ≥ M we have(
λi −1 1λ <N )`(λ)
XY vi − qvj Y
i
v
−
s
1
i
vi
,
F•λ (v1 , . . . , vk ) = C(λ)
σ
v
1
−
sv
1
−
sv
i − vj i=1iiσ∈S1≤i<j≤kk(2.78)
where the symmetric group Sk acts on the variables vi but not on elements of the signature
λi , and the constant prefactor has the formN
(1 − q)k Y (s2 ; q)mi (λ)C(λ) =.
(q; q)k−`(λ) i=1 (q; q)mi (λ)(2.79)

Proof of theorem 2.6.6. We make use of the symmetrization formula (2.78) (recall that
we have replaced the parameter q by t throughout this subsection). We use the notationYvl − tvrA=
vl − vr
1≤l<r≤M

and

B=

`(λ)
Y
r=1

55

vr



vr − s1 − svrλr −1 

1
1 − svr

1λr <N,so that F•λ = C(λ)

P∗
σ∈SM σ{AB}. The operator D1,N acts asD•1,N F•λ = C(λ)

M Y
MXvi − tvj X
i=1 j=1j6=ivi − vj σ∈S

Ci,N (σ{AB}).

M

The action of Ci,N on the product σ{AB} can be split as
Ci,N (σ{AB}) = lim σ{A}ε→0vi =1/ε× Ci,N (σ{B}).

(2.80)

Assume now that λ0N = L, that is λ1 = · · · = λL = N and λL+1 < N , for someL ∈ {0, . . . , M }. We focus on the second factor of (2.80). A simple computation showsthat(
σ{B} if i ∈ σ({1, . . . , L}),
Ci,N (σ{B}) =(2.81)
0else,that in particular, implies that Ci,N σ{B} = 0 when L = 0, confirming (2.77) in thisspecific case.
For L > 0 and a permutation σ such that i ∈ σ({1, . . . , L}), call k̄ the element suchthat σ(k̄) = i. We rewrite A into a product of factors A = A1 A2 A3 , obtained dividing
the triangular product as
Y vl − tvk̄Y
Y vl − tvr
vl − tvr
,
A2 =
,
A3 =
.
A1 =
vl − vr
vl − vk̄v
l − vr1≤l<M
1≤l<k̄

1≤l<r<k̄

max(l,k̄)<r≤M

We can evaluate the first factor in the right-hand site of (2.80) asMYvi − tvj

e2 A3 },
lim σ{A}= tk̄−1 σ{A1 A
ε→0
v
−
v
v
=1/ε
i
j
j=1 i
j6=i

where
e2 :=
A

Y vk̄ − tvl

1≤l<k̄

vk̄ − vl.

The action of D∗1,N on the sHL function can be therefore expressed (ignoring C(λ)) as
L
X
k̄=1

k̄−1

t

M X
X

i=1 σ∈SMσ(k̄)=ie2 A3 B} =σ{A1 A

L
X
k̄=1

tk̄−1

X

σ∈SM

e2 A3 B}.
σ{A1 A(2.82)

e2 A3 B} is equal to one of the
To prove relation (2.77) we show that each term σ{A1 A
terms τ {A1 A2 A3 B} in the expansion of the original sHL function. For each permutation
τ ∈ SM and each k̄ define the permutation σ asτ (j + 1) if j = 1, . . . , k̄ − 1,
σ(j) = τ (1)if j = k̄,


τ (j)
if j = k̄ + 1, . . . M.
56

With this choice we can easily check that σ{A3 B} = τ {A3 B} and more crucially thate2 } = τ {A1 A2 } since the cyclic shift in the first k̄ terms of σ makes up for theσ{A1 Ae2 and A2 . This in particular shows that the symmetric sum in the rightexchange of A
hand side of (2.82) is independentomitted,P of k̄ and it is equal, up to a factor C(λ) that we0
to F•λ (v1 , . . . , vM ). The sum Lk̄=1 tk̄−1 is the desired eigenvalue e1 (1, t, . . . , tλN −1 ). This
completes the proof.
Remark 2.6.8 (Limit to the Hall–Littlewood case). In the limit s → 0, the new operator
D•1,N (2.76) acting on the dual sHL functions should be replaced by
∗
D1,N
=

M Y
M
X
vj − tvl
j=1 l=1l6=jvj − vl

vjN lim εN Tε−1 ,vj ,ε→0(2.83)∗
by mimicking the action (2.81). Similarly to Theorem 2.6.6, one can show that D1,N
acts
diagonally on the Hall–Littlewood polynomials Pλ (·; 0, t).
The same operator (2.83) can be also obtained as a q → 0 limit of a certain operator
diagonal in the Macdonald polynomials Pλ (·; q, t). Take the first Macdonald q −1 -difference
operator
M Y
MXtxi − xj
Tq−1 ,xj .
(2.84)
M1 =
x
−
x
i
j
j=1 i=1i6=jP−λi i−1t
It acts on the Macdonald polynomials Pλ (x1 , . . . , xM ; q, t) with eigenvalues M
i=1 q(this follows from, e.g., [BC14, Section 2.2.3]). Denote by PN the subspace of polynomials
in x1 , . . . , xM which have degree ≤ N in each of the variables xi . It is spanned by the
Macdonald polynomials Pλ (x1 , . . . , xM ; q, t) with λ1 ≤ N , i.e., λ ⊆ Box(N, M ). On PN
consider the operator q N M1 . Its limit as q → 0 is well-defined. By looking at eigenvalues
on Hall–Littlewood polynomials Pλ (x1 , . . . , xM ; 0, t) with λ1 ≤ N , one readily sees that∗.
this limit coincides with D1,N

2.6.3

Eigenrelations for the spin q-Whittaker polynomials

The duality between sHL functions and sqW polynomials (Corollary 2.3.11 and Proposition 2.2.3) allows to pass from the eigenoperators for the sHL functions to the ones forthe sqW polynomials.
Definition 2.6.9 (Spin q-Whittaker difference operators). Fix a positive integer N , anddefine the s-deformed q-Whittaker operators
N Y
NX(1 + sxi )Tq,xi ,

(2.85)N Y
N
X(1 + s/xi )Tq−1 ,xi .

(2.86)

D1 :=1 − xi /xj
i=1 j=1
j6=i

and
D1 :=

i=1 j=1j6=i

1 − xj /xi
57Let us make a remarks after this definition.
Remark 2.6.10. The operators D1 and D1 reduce for s = 0 to the t = 0 specializationsof the two Macdonald
The first one is the standard first order
PNq-difference
Q txi −xoperators.
j1
Macdonald operator i=1 j6=i xi −xj Tq,xi (denoted by DN
in [Mac95, Ch. VI]), and the
PN Q xi −txj1
second one is i=1 j6=i xi −xj Tq−1 ,xi (denoted by D̃N in [BC14, Section 2.2.3]).
We establish two eigenrelations for the sqW polynomials in the next two theorems.

Theorem 2.6.11. For all signatures λ ∈ SignN we haveD1 Fλ (x1 , . . . , xN ) = q λN Fλ (x1 , . . . , xN ).

(2.87)

Proof. We will prove the identity
1 − (1 − q)D∗1,N Π(x; v) = D1 Π(x; v),whereΠ(x; v) =

M 
Y
j=1

1
1 − svj

N −1(2.88)N Y
M
Y×(1 + vj xi ).

(2.89)

i=1 j=1

Indeed, modulo (2.88), the Cauchy Identity (2.27) and the eigenrelations (2.77) imply
X
X
q λN Fλ (x1 , . . . , xN )F•λ0 (v1 , . . . , vM ) =D1 Fλ (x1 , . . . , xN )F•λ0 (v1 , . . . , vM ),λ⊆Box(N,M )λ⊆Box(N,M )and hence (2.87) follows by orthogonality of the sHL functions (Proposition 2.2.3).
It thus suffices to establish (2.88). DefineMYh(z) :=(1 + vj z).
j=1

We haveD1 Π(x; v)
D1 h(x1 ) · · · h(xN )==−Π(x; v)h(x1 ) · · · h(xN )INY
xi (1 + sz) h(qz)x1 ,...,xN i=1

xi − zdz,
h(z) z(1 + sz)

where in the second equality we used the residue expansion of the complex integral andthe contour encircles only the poles x1 , . . . , xN . By subtracting 1 from both sides, we
can enlarge the complex contour to also include the pole at z = 0 (note that h(z) is
nonsingular at z = 0). After a change of variable z = −1/w, we get(−1 + D1 )Π(x; v)=−
Π(x; v)IMY
w − qvk

v1 ,...,vM k=1

w − vk58(w − s)

N −1

N
Y

xj
dw.
1 + xj w
j=1(2.90)

In the right-hand side of (2.90), after the change of variable, we switched the contour toa positively oriented curve around v1 , . . . , vM , which yielded the negative sign in front.
Using
!
N
Y
1
x1 · · · xNlim ε(1 + xj /ε) =N−1ε→0(1 − s/ε)(−s)N −1j=1
and expanding the right-hand side of (2.90) as a sum of residues, we can rewrite it as−(1 − q)D∗1,N Π(x; v).
Π(x; v)

This proves (2.88), and hence the desired eigenrelation (2.87).
Theorem 2.6.12. For all signatures λ ∈ SignN we haveD1 Fλ = q −λ1 Fλ .

(2.91)

Proof. The proof of this eigenrelation is identical to that of theorem 2.6.11. And it uses
the fact that


∗
−M
q
1 − (1 − q)D1 Π(x; v) = D1 Π(x; v),
where Π(x; v) is given by (2.89). We will not repeat the detailed argument here.

2.6.4

Commutation and conjugation

The q-difference operators D1 (2.85) and D1 (2.86) commute. For this statement we cannot appeal to the eigenrelations of Theorems 2.6.11 and 2.6.12 since we did not prove thatthe sqW polynomials form a basis for the ring of symmetric polynomials in N variables.
Nevertheless, the commutation can be checked independently:
Proposition 2.6.13. We have D1 D1 F = D1 D1 F for all symmetric polynomials F in N
variables.
Proof. By polarization, it suffices to check the action on product form functions F (x1 , . . . , xN ) =f (x1 ) . . . f (xN ), where f (x) is an arbitrary polynomial.
The action of each operator can we written as a contour integral:

I Y
N 
xi (1 + sz) f (qz)dz
1f (xi ),
D1 F = −2πi i=1
xi − zf (z) z(1 + sz)

I Y
N 
1
w + s f (q −1 w) dw
D1 F =f (xi ),
2πi i=1
w − xif (w) w + s
where both integrals are over a contour containing x1 , . . . , xN and no other poles of theintegrand. Throughout the proof we assume that all contours exist, which might impose
some restrictions on the xi ’s. After checking the commutation under the restrictions, we
can lift these restrictions by an analytic continuation.
59

We have1
D1 D1 F = −(2πi)2N 
Ydz
w + s xi (1 + sz) w − z f (qz)f (q −1 w) dw,
f (xi )
w − xi xi − z
w − qz f (z)f (w) w + s z(1 + sz)
1
γw
i=1

I I
γz1

where γz1 contains both γw1 and q −1 γw1 , while γw1 is around x1 , . . . , xN and no other poles.
In the other order, we have1
D1 D1 F = −(2πi)2I YN xi (1 + sz) w + s q −1 (w − z) f (q −1 w)f (qz)dzdw
f (xi ),
−1x i − z w − xiq w−z
f (w)f (z) z(1 + sz) w + s2γw
γz2 i=1I

but now γw2 contains both γz2 and qγz2 , while γz2 is around x1 , . . . , xN and no other poles.
Note that the integrands in both formulas coincide.
In the first expression, deform the integration contour γz1 to coincide with γw1 , whichpicks up the residue at z = q −1 w. In the second expression, deform the contour γw2 tocoincide with γz2 , which picks up the residue at w = qz. The resulting double contour
integrals are over the same contours and are thus equal. It remains to check the equalityof the single integrals of the residues. We haveRes(integrand in D1 D1 ) = −(−1)N (1 − q)(s + w)N −1 (q + sw)N −1
−1

z=q

w

N −1

N

Res (integrand in D1 D1 ) = (−1) (1 − q)(1 + sz)

w=qz

N −1

(s + qz)

NY

xi f (xi )
,
(w
−
x
)(w
−
qx
)
i
i
i=1

N
Y

xi f (xi )(z − xi )(qz − xi )i=1

We must show that the integral of the first expression over γw1 is the same as the integral of
the second expression over γz2 . Noting that both expressions have zero residue at infinity
due to quadratic decay, we can compute the first integral as a sum of minus residues at
w = qxi . Then one readily sees that each minus residue at w = qxi is the same as theresidue of the second expression at z = xi . This shows the desired commutation.
The discussion in the rest of this subsection aims in part to demonstrate why the
result of Proposition 2.6.13 is a rather unexpected one.
Both operators D1 (2.85) and D1 (2.86) are related via conjugation to q-Whittakerdifference operators. The latter are t = 0 degenerations of the Macdonald q-difference
operators from [Mac95]. Denote for r = 1, . . . , N :WNr :=

X Y|I|=r i∈I, j ∈I/Y
1Tq,xi ,
1 − xi /xj i∈I

W̃Nr :=X Y
I

i∈I, j ∈I/Y1
Tq−1 ,xi , (2.92)
1 − xj /xi i∈I

where the sums are over subsets of {1, . . . , N } of cardinality r. These operators are
diagonal in the usual q-Whittaker polynomials (which are t = 0 versions of the Macdonald
polynomials). In particular, WN1 and W̃N1 have eigenvalues q λN and q −λ1 , respectively, on

60

q-Whittaker polynomials. All the operators WNr , W̃Nr , r = 1, . . . , N , commute. We refer
to Sections 2.2.2 and 3.1.3 in [BC14] for details. Let
UN :=

N
Y

1
,
N −1
(−sx
;
q)
i
∞
i=1

VN :=

N
Y

1
.
N −1
(−s/x
;
q)
i
∞
i=1

A straightforward computation shows:
Proposition 2.6.14. The spin q-Whittaker operators (2.85), (2.86) are conjugates of thefirst q-Whittaker operators (2.92):
1
D1 = U−1
N WN UN ,1
D1 = V−1
N W̃N VN ,where UN , etc., mean multiplication operators.
Because the q-Whittaker operators (2.92) commute, we get many operators commuting
with either D1 or D1 . That is, for r = 1, . . . , N we have:




r
r
D1 , V−1
D1 , U−1(2.93)
N W̃N VN = 0.
N WN UN = 0,

For example,

2
U−1
N WN UN =X(1 + sxi )N −1 (1 + sxk )N −1

i,k : i6=kY

1
Tq,xi Tq,xk .
(1 − xi /xj )(1 − xk /xj )j : j6=i,k(2.94)
2
However, one can directly check that the operator U−1
W
U
does
not
commute
with
D1 .
N N
N
−1
−1
r
r
This suggests that the operators UN WN UN or VN W̃N VN , r ≥ 2, should not be diagonalin the spin q-Whittaker polynomials Fλ . The following example shows that this is indeed
the case:
Example 2.6.15. Take N = 2, then (1 − s2 )F(1,0) (x1 , x2 ) = s + x1 + x2 + sx1 x2 . Applying(2.94) to this function, we obtain (1 + sx1 )(1 + sx2 )(s + qx1 + qx2 + sq 2 x1 x2 ), which is
not proportional to F(1,0) (x1 , x2 ) unless s = 0. Note that for s = 0 both UN and VN are
the same (and are equal to the identity), and D1 , D1 are the usual q-Whittaker difference
operators.
r
We also observe that by (2.93), polynomials of the form U−1
N WN UN Fλ , r = 2, . . . , N ,rare eigenfunctions of the operator D1 with eigenvalues q λN . Similarly, V−1
N W̃N VN Fλ are−λ1
eigenfunctions of D1 with eigenvalues q . However, one can check that D1 does not act
2
diagonally on, say, the polynomial U−1
N WN UN F(1,0) .

It remains unclear how to construct higher order q-difference operators which would
be diagonal in the sqW polynomials (and whether such operators exist at all).

61

2.7

Open problems

2.7.1

Orthogonality and spectral theory for spin q-Whittakerpolynomials

The q-Whittaker polynomials satisfy orthogonality relations coming from (the t = 0
degeneration of) the Macdonald torus scalar product [Mac95, Ch. VI.9]. This relation
states that the s = 0 versions of Fλ (z1 , . . . , zN ) and Fµ (1/z1 , . . . , 1/zN ) are orthogonal
to each other when µ 6= λ with respect to a certain weight on the N -dimensional torus
TN = {|zi | = 1, i = 1, . . . , N }.
Remark 2.7.1. Under the generalization with a spin parameter, the spin Hall–Littlewood
polynomials also satisfy a version of the torus orthogonality (called spatial orthogonality in[Bor+15b, Corollary 3.10], see also [Bor17], [BMP19, Proposition 8.6]), as well as anotherbiorthogonality involving the summation of over λ instead of integration over z. Here we
discuss only the former conjectural orthogonality of the spin q-Whittaker polynomials.
DefinemNq,s (z1 , . . . , zN ) :=NY
(s2 , zi /zj ; q)∞ Y 1
1,
N ! 1≤i6=j≤N (−szi , −s/zi ; q)∞ i=1 2πizi(z1 , . . . , zN ) ∈ TN .

When s = 0, mN
q,s reduces to the orthogonality measure of the q-Whittaker polynoN
mials on T , which is a t = 0 degeneration of the Macdonald’s torus scalar product∆(z1 , . . . , zN ; q, t), cf. [Mac95, VI.(9.2)].
Lemma 2.7.2. Both eigenoperators D1 , D1 (2.85), (2.86) for the spin q-Whittaker polynomials are self-adjoint with respect to the scalar productZf (z1 , . . . , zN ) g(z1 , . . . , zN ) mN
hf, giq,s :=q,s (z1 , . . . , zN ) dz1 . . . dzN ,TN

where f, g are Laurent polynomials with coefficients in R(q, s).
Proof. A direct verification.
Conjecture 2.7.3. We have for all signatures λ, µ:ZFλ (z1 , . . . , zN ) Fµ (1/z1 , . . . , 1/zN ) mNq,s (z1 , . . . , zN ) dz1 · · · dzN = cλ 1λ=µ ,(2.95)

TN

where
cλ =

N
−1
Y

(s2 ; q)∞ (q; q)λi −λi+1.
2 ; q)
(q;
q)
(s
∞
λ
−λ
i
i+1
i=1

(2.96)

Note that for N ≤ 2 the statement (up to the concrete formula for the norm cλ ) follows
from Lemma 2.7.2 and the eigenrelations of Theorems 2.6.11 and 2.6.12. However, forN ≥ 3 the two operators D1 , D1 are not sufficient to conclude orthogonality.
62Remark 2.7.4. When s = 0, the constant cλ (2.96) coincides with the t = 0 degenerationof the torus scalar product norm of a Macdonald polynomial [Mac95, Ch. VI.9, Example
1].
Let us present one further argument in favor of Conjecture 2.7.3. It was proven in[IMS20, Proposition 4.10] that the probability mass function of a tagged particle in thehomogeneous q-Hahn Tasep with parameters ν = s2 and µ = −ys isN −1 ZtN YΠ(zj ; y)(q; q)∞
N
mq,s (z1 , . . . , zN )P(H(N, t) = `) =(s2 ; q)∞Π(−s; y)TN
j=1
×where Π(z; y) =



(−s)N
z1 · · · zN

`

N

−1
( z(−s); q)∞ (s2 ; q)N∞
1 ···zN(−sz1 , . . . , −szN ; q)∞dz1 · · · dzN ,(2.97)(−sz; q)∞. The same probability can be expressed as(−sy; q)∞XF∗λ (y, . . . , y)Fλ (−s, . . . , −s),Π(−s; y)λ∈SignN

λN =`Assuming Conjecture 2.7.3, this sum becomes
t 
` X
Z
N 
Y
Π(zj ; y)(−s)N(−s)|λ|NFλ (1/z1 , . . . , 1/zN ) dz1 · · · dzN .
mq,s (z1 , . . . , zN )
Π(−s;
y)
z
·
·
·
z
c
1
N
λ
TN
j=1
λ∈SignN

λN =0Here we used the Cauchy Identity and the torus scalar product to express the dual function
F∗λ , and the shifting rule of Proposition 2.3.7 to take out the monomial of degree `.
Evaluating the sum inside the integral as in (2.23), we recover exactly (2.97).

2.7.2

Accessing spin q-Whittaker polynomials via free field

In this thesis we did not focus on Fredholm determinantal structures for marginals of
spin q-Whittaker processes. These aspects have been covered quite extensively in literature for specific models in the last few years [Cor14], [CP16], [IS19], [IMS20], [BMP19].
Techniques to access such Fredholm determinantal formulas usually rely on manipulations
with integral representations of q-moments (as in [BC14], [BCS14]).
In the realm of Macdonald processes there exists an alternative approach to expose
the determinantal nature of specific observables. This is done via a free field realization of
Macdonald functions and Macdonald operators [Fei+09], [Kos19], see also [Bor+16] and,e.g., [FW09] for the Hall–Littlewood case. In the yet simpler case of the Schur processes
this reduces to the infinite wedge representation of Schur functions [Oko01], [OR03]. It
would be of great interest to understand to what extent our spin q-Whittaker functions,operators, and processes admit a description in terms of Fock type representations of ahypothetical (q, s)-deformed Heisenberg algebra.
63Chapter 3
Integrable random fields of Youngdiagrams
This chapter presents results contained in sections 6,7,9 of[BMP19] A. Bufetov, M. Mucciconi, and L. Petrov. “Yang-Baxter random fields and stochastic vertex models”. In: arXiv preprint (2019). arXiv:1905.06815 [math.PR]. To
appear in Adv. Math.
and of sections 4,5,6 of[MP20] M. Mucciconi and L. Petrov. “Spin q-Whitaker polynomials and deformed quantumToda”. In: arXiv preprint (2020). arXiv:2003.14260 [math.PR]

3.1

Integrable stochastic dynamics on interlacing arrays

In this section we implement the general scheme of passing from symmetric functions
satisfying Cauchy type summation identities to probability measures. This approachclosely follows the ideas of Schur / Macdonald processes [OR03], [BC14]. We use theframework of skew Cauchy structures which is explained in detail in [BMP19, Section 2].

3.1.1

Skew Cauchy structures and random fields

We say that two families of functions F, G form a skew Cauchy structure if they satisfy
the following properties:
1. Fλ/µ , Gλ/µ are symmetric rational functions in their respective variables, parametrizedby pairs of Young diagrams λ/µ (or signatures with appropriate numbers of parts).
In particular, Fλ/µ , Gλ/µ are nonzero only if µ ⊆ λ.

64

2. Branching rules: for all µ, λ we haveX
Fν/λ (u1 , . . . , un ) =Fµ/λ (u1 , . . . , un−1 )Fν/µ (un )µfor any n and any set of variables u1 , . . . , un , and analogously for G.
3. There exists a function Π and a set Adm ⊆ C2 such that the skew Cauchy identityXX
Π(u; v)Fµ/κ (u)Gλ/κ (v) =Fν/λ (u) Gν/µ (v)
(3.1)
κ

ν

holds numerically for all (u, v) ∈ Adm. Note that u, v stand for single variables, asin Theorems 2.2.5, 2.3.10 and 2.3.15.

4. There exist two sets P, Ṗ ⊆ C, with P × Ṗ ⊆ Adm, such that for any choice ofu ∈ P and v ∈ Ṗ the functions Fλ/µ (u), Gλ/µ (v) are non negative for all λ, µ. In thiscase we say that u, v are positive specializations. (Nonnegativity of single-variable
functions together with branching implies nonnegativity of multi-variable versionsof the functions.)
Consider now two sequences of Young diagrams (or signatures) ~λ = (λ1 , . . . , λn ) andµ~ = (µ1 , . . . , µn−1 ) withλ1 ⊇ µ1 ⊆ λ2 ⊇ µ2 ⊆ · · · µn−1 ⊆ λn ,and sequences of positive specializations u1 , . . . , un and v1 , . . . , vn respectively of F and
G. The F/G process is the probability measure
!
n−1
Y
1
Gλi /µi (vi )Fλi+1 /µi (ui+1 ) Gλn (vn ),(3.2)
Prob(~λ, µ~ ) = Fλ1 (u1 )Z
i=1Qwhere the normalization constant is Z = i,j Π(ui ; vj ).
For applications to stochastic dynamics, it is of interest to consider random fields{λ } of Young diagrams indexed by Z2≥0 , whose marginal distributions along downright paths are given by suitable F/G processes. A down-right path is(i,j)$ = {$k = (ik , jk ) : 0 ≤ k ≤ L}, where i0 = jL = 0 and $k+1 − $k ∈ {e1 , −e2 }.
Here L is arbitrary and depends on $, and e1 , e2 are the standard basis vectors (1, 0), (0, 1).
Definition 3.1.1. Consider positive specializations u1 , u2 , . . . and v1 , v2 , . . . respectivelyof functions F and G. An F/G field is a probability measure on the set {λ(i,j) : i, j ∈ Z≥0 }that associates the probabilityYY
1
Gλ$k /λ$k+1 (vjk ).
(3.3)
Fλ$k+1 /λ$k (uik+1 )Z$ k : $ =$ +e
k: $
=$ −e
k+1

k

1

k+1

k

2

to the event of finding Young diagramsQλ$1 , . . . , λ$L−1 along an down-right path $. Herethe normalization constant is Z$ = (i,j) below $ Π(ui ; vj ), and at the boundary we fix
λ(0,j) = λ(i,0) = ∅ with probability one.
65

Remark 3.1.2. While the F/G process is defined uniquely by (3.2), it is not determineduniquely by Definition 3.1.1. Below in this section we outline two different constructionsof a field in our particular cases. See also the discussion in [BMP19, Section 2.6] for moredetails and additional references.
To visualize an F/G field, decorate edges (i − 1, j) → (i, j) of the first quadrant withspecializations ui , and edges (i, j − 1) → (i, j) with vj . Then for each down-right path $,the probability of finding the sequence λ$k is computed by climbing down $ and picking
up skew functions F(uik ) along horizontal edges, and G(vjk ) along vertical edges. See
Figure 3.1 for an illustration.
jGv4

λ(1,3) λ(2,3) λ(3,3) λ(4,3)
v3
λ(1,2) λ(2,2) λ(3,2) λ(4,2)
v2
λ(1,1) λ(2,1) λ(3,1) λ(4,1)
v1

i

$
u1

u2

u3

u4

u5

F

Figure 3.1: A down-right path (highlighted) in a random field, and edge decorations.
In this thesis, particularly interesting instances of F/G processes will be those arisingwhen considering paths $ of the form (0, T ) → (N, T ) → (N, 0). Taking the marginal
distribution of λ(N,1) , . . . , λ(N,T ) , we arrive at the following definition:
Definition 3.1.3. The ascending F/G process is the probability measure on the set ofYoung diagrams (or signatures)λ1 ⊆ λ2 ⊆ · · · ⊆ λN ,assigning to each such sequence the probability weight
QN QT
i=1

3.1.2

1

j=1 Π(ui ; vj )Fλ1 (u1 ) Fλ2 /λ1 (u2 ) · · · FλN /λN −1 (uN ) GλN (v1 , · · · , vT ).

Fields based on spin Hall-Littlewood and spin q-Whittakerpolynomials

Here we specialize skew Cauchy structures to three cases involving spin q-Whittaker andspin Hall–Littlewood functions.

66

Definition 3.1.4 (sHL/sHL field). Let s ∈ (−1, 0) and take parameters ui , vj ∈ [0, 1).
The sHL/sHL field is obtained by specializing Fλ/µ (xi ) = Fλ0 /µ0 (xi ) and Gλ/µ (vj ) =F∗λ0 /µ0 (vj ).
The corresponding skew Cauchy identity is given in Theorem 2.2.5. One readily verifies
that the sHL functions specialized like this are nonnegative, which leads to probability distributions. Joint distributions along down-right paths in this field are given by sHL/sHL
processes which are specializations of (3.2).
Definition 3.1.5 (sqW/sHL field). Let s ∈ (−1, 0) and take parameters xi ∈ [−s, −s−1 ],vj ∈ [0, 1). The sqW/sHL field is obtained by specializing Fλ/µ (xi ) = Fλ/µ (xi ) andGλ/µ (vj ) = F•λ0 /µ0 (vj ).
The corresponding skew Cauchy identity is given in Theorem 2.3.10. One readily
verifies that the sHL and sqW functions specialized like this are nonnegative, which leads
to probability distributions. Joint distributions along down-right paths in this field are
given by sqW/sHL processes which are specializations of (3.2).
Definition 3.1.6 (sqW/sqW field). Let s ∈ (−1, 0) and take parameters xi , yj ∈ [−s, −s−1 ].
The sqW/sqW field is obtained by specializing Fλ/µ (xi ) = Fλ/µ (xi ) and Gλ/µ (yj ) =F∗λ/µ (yj ).
The corresponding skew Cauchy identity is Theorem 2.3.15. The range of parameters here also leads to nonnegative functions F, F∗ , thus producing probability measures.
Joint distributions along down-right paths in the sqW/sqW field are given by sqW/sqW
processes which are specializations of (3.2).

3.1.3

Sampling a field via bijectivization

As mentioned in Remark 3.1.2, a random field is not determined uniquely. Moreover, its
properties (like marginal stochastic dynamics) heavily rely on a particular choice of the
field’s construction. This choice can be encoded by certain Markov transition operators.
Let us return to the general formalism of skew Cauchy structures.
Suppose that we have Markov transition operators
Ufwd
u,v (κ → ν | λ, µ)

and

Ubwd
u,v (ν → κ | λ, µ),that satisfy the reversibility condition
bwd
Ufwd
u,v (κ → ν | λ, µ)Π(u; v)Fµ/κ (u)Gλ/κ (v) = Uu,v (ν → κ | λ, µ)Fν/λ (u) Gν/µ (v).

(3.4)

Here Ufwd
u,v (κ → ν | λ, µ) encodes the probability of a transition κ → ν conditioned on λ, µ,whereas Ubwd
u,v (ν → κ | λ, µ) describes the probability of the opposite move (specializationsu, v are assumed positive). See Figure 1.3, left, for an illustration. Summing (3.4) overboth ν and κ and using the Markov property of Ufwd , Ubwd , one recovers the skew Cauchy
Identity (3.1). Condition (3.4) determines Ubwd once Ufwd if given, and vice versa.
If Ufwd is given, we can construct a random field {λ(i,j) : i, j ∈ Z≥0 } as in fig. 1.3,
right panel. Namely, fix empty boundary conditions. Inductively for n ≥ 2, assuming we
67

0

0

already sampled Young diagrams λ(i,j) with i + j ≤ n, pick λ(i ,j ) for each i0 + j 0 = n + 1
at random with probabilities
0

0

0

0

0

0

0

0

(i −1,j −1)
Ufwd→ λ(i ,j ) | λ(i −1,j ) , λ(i ,j −1) ),ui0 ,vj 0 (λindependently for various pairs (i0 , j 0 ). We say that the field is generated by Ufwd .
Proposition 3.1.7. Assume that Ufwd is known. Then the procedure described right above
samples an F/G field.
Proof. One has to show that the distribution of the Young diagrams along any down-rightpath is described by the corresponding F/G process. This is readily verified by induction
on adding one box to the area below the down-right path, and using (3.4). We omit thedetails.

3.1.4

Borodin–Ferrari fields

Let us now describe a particular choice of the forward transition probabilities whichguarantees the existence of a field for a skew Cauchy structure. This construction isbased on [BF14] and follows an earlier coupling idea of [DF90]. ChooseFν/λ (u)Gν/µ (v)P,
Π(u; v) κ Fµ/κ (u)Gλ/κ (v)Π(u; v)Fµ/κ (u)Gλ/κ (v)P.
Ubwd
u,v (ν → κ | λ, µ) =
ν Fν/λ (u)Gν/µ (v)Ufwdu,v (κ → ν | λ, µ) =(3.5)

In general, although transition probabilities (3.5) are explicit, in particular examples theirconcrete meaning may be far from transparent.
A helpful simplification can be made if we assume that G admits expansionGν/µ (v) = (v − v ∗ )d(ν/µ) (gν/µ + O(v − v ∗ )),(3.6)

for some fixed value v ∗ independent of ν, µ, coefficients gν/µ , and a “nice” degree function dsuch that d(ν/ν) = 0. Then one can consider a Poisson-type scaling limit of the field (3.5)
as vj → v ∗ for all j. Under this scaling, the discrete vertical axis becomes continuous,and the field turns into a Markov dynamics {λ(i,t) : i ∈ Z≥0 , t ∈ R≥0 }, where t is thecontinuous time variable. The dynamics lives on sequences of signatures.
When F, G are Schur functions, such continuous processes is the push–block dynamics
introduced in [BF14].

3.1.5

Bijectivization of the Yang–Baxter equation

In many cases, skew Cauchy Identities descend directly from the Yang–Baxter equation
and we have seen examples of this phenomenon in chapter 2. This observation was usedin [BP19], [BMP19] to provide an explicit construction of random fields for sHL and sqW
68

functions, which we briefly recall here. In general, this approach produces fields which
differ from the Borodin–Ferrari ones. On the other hand, Yang–Baxter fields by design
possess Markovian marginals.
For any given identity with positive terms
XX
w(a) =
w(b),
(3.7)
a∈A

b∈B

we say that two stochastic matrices pfwd (a, b) and pbwd (b, a) (with indices a ∈ A, b ∈ B)
form a (stochastic) bijectivization of identity (3.7) if they satisfy the reversibility condition
pfwd (a → b) w(a) = pbwd (b → a) w(b)for all a ∈ A, b ∈ B.

A bijectivization always exists since we can take pfwd (a → b) ∝ w(b). A bijectivization is
unique only when A or B has a single element. Another simple case is given when both
A and B have only two elements.
Example 3.1.8. When A = {a1 , a2 } and B = {b1 , b2 }, identity (3.7) becomesw(a1 ) + w(a2 ) = w(b1 ) + w(b2 ).
In this case all stochastic bijectivizations pfwd , pbwd are expressed as
pfwd (a1 → b1 ) = γ,w(a2 ) − w(b2 ) + (1 − γ)w(a1 ),w(a2 )
pfwd (a2 → b2 ) = 1 − pfwd (a2 → b1 ),pfwd (a2 → b1 ) =pfwd (a1 → b2 ) = 1 − γ,
for a parameter γ ∈ [0, 1].

Let now (3.7) be one of the Yang–Baxter equations from Chapter B, corresponding to
fig. B.5 (b) or fig. B.7 (b). Let us rewrite them in a unified notation asX
X
wr(K 0 | I, J),(3.8)
wl(K | I, J) =K0

Kwhere I = {i1 , i2 , i3 }, J = {j1 , j2 , j3 }, K = {k1 , k2 , k3 } and K 0 = {k10 , k20 , k30 }, and weightfunctions wl, wr denote the terms in the left and right-hand sides of each of the YangBaxter equations of chapter B involving pairs of non-dual and dual weights. Equations(B.26) and (B.27) involving corner weights, by agreement, correspond to the choice j1 = ∅.
0bwd
0Denote by pfwd
I,J (K → K ) and pI,J (K → K) a stochastic bijectivization of (3.8).
fwd
Then pI,J is the probability of moving the cross from left to right (in the local configuration in Figure B.5 (b)), while transforming the occupation numbers K into K 0 . The
0probabilities pbwd
I,J (K → K) similarly correspond to moving the cross from right to left.
By the conservation of paths at each vertex, once I, J are fixed, the configuration K iscompletely determined specifying only one of the numbers k1 , k2 , or k3 (and similarly forK 0 ).
69

Bijectivizations of the Yang–Baxter equation are building blocks of operators Ufwd , Ubwd .
Given Young diagrams κ, µ, λ, ν we identify path configurations through two rows ofvertices as in Figure 3.2, that is using their dual path representation1 as discussed in
Section 2.1.2. Vertices crossed by blue paths are assigned non dual weights (w or W )(2.12)–(2.14) whereas those in red have dual weights (w∗ or W ∗ ). We assume that at theleftmost column an infinite number of paths flows in the vertical direction. The transition
probability Ufwd (κ → ν | λ, µ) is the product of probabilities of sequential local moves
pfwd obtained dragging the cross vertex from the leftmost column to the right. In case weare considering transition probabilities between signatures such dragging will terminate ata specific location N , where the corner weights W will necessarily have to be employed.
The operator Ubwd is constructed using the opposite local moves with probabilities
pbwd , starting from a column very far to the right. See Figure 3.2 for an illustration.
Definition 3.1.9. The random field constructed from operators Ufwd takes the name of
Yang-Baxter field. In this case we say that Ufwd is generated by local moves pfwd .

3.1.6

Three Yang-Baxter fields

In this subsection we specialize the construction described in section 3.1.5 to demonstratethat Yang-Baxter fields sample random fields weighted by sHL and sqW functions.
We consider first a stochastic bijectivization pfwd and pbwd of the Yang-Baxter equation(B.2). We fix parameters q ∈ (0, 1), s ∈ (−1, 0), and ui , vj ∈ [0, 1), i, j ∈ Z≥1 . To
distinguish this case from the one that we will follow we will denote the Markov operators
bwd
generated by pfwd and pbwd respectively as UfwdsHL(u),sHL(v) and UsHL(u),sHL(v) .
Proposition 3.1.10. The random field {λ(i,j) : i, j ∈ Z≥0 } generated by UfwdsHL(ui ),sHL(vj ) isa sHL/sHL field as in definition 3.1.4. In particular the single-point distributions havethe form
Y 1 − uj 0 vi0Fν 0 (u1 , . . . , ui ) F∗ν 0 (v1 , . . . , vj ),Prob λ(i,j) = ν =0 v i01−quj01≤i ≤i1≤j 0 ≤jwhere ν is an arbitrary fixed Young diagram.
The proof of proposition 3.1.10 is a straightforward consequence of the following bijective proof of the skew Cauchy identity (2.11).
Proposition 3.1.11. For any four Young diagrams µ, κ, λ, ν we have
1 − quv fwd
U(κ → ν | λ, µ) Fλ0 /κ0 (u) F∗µ0 /κ0 (v)
1 − uv sHL(u),sHL(v)∗= UbwdsHL(u),sHL(v) (ν → κ | λ, µ) Fν 0 /λ0 (v) Fν 0 /µ0 (u). (3.9)

Summing (3.9) over both κ and ν, we obtain the skew Cauchy identity for the stable sHL
functions (Theorem 2.2.5).
to a Young diagram λ we associate the arrow representation |λ0 i. We switch to this convention inorder to treat random fields corresponding to sHL and sqW functions on an equal footing170µc−1 −µc

µc −µc+1µc+1 − µc+2

νc−1 −νcκc −κc+1

κc+1 −κc+2

λc−1 −λc

λc −λc+1

λc+1 − λc+2

pfwd

µc−1 −µcµc −µc+1µc+2 − µc+2

νc−1 −νcνc − νc+1

κc+1 −κc+2

λc−1 −λc

λc −λc+1

λc+1 − λc+2

pbwd

Figure 3.2: A local random move in a Yang–Baxter field. Moving the cross through thecolumn c updates the value of κc − κc+1 to νc − νc+1 .
Proof. The product Fλ0 /κ0 (u)F∗µ0 /κ0 (v) is the weight of a configuration µ  κ ≺ λ in a∗-weighted
vertex model obtained attaching a wu,s -weighted row of vertices on top of a wv,srow of vertices. The leftmost column is occupied by infinitely many paths. The factor(1 − quv)/(1 − uv) is the Ruv weight of a crossattached at the left of the lattice.
fwd
Now we employ the definition of UsHL(u),sHL(v) and drag the cross all the way to the right,
replacing κ by the random ν. This procedure, along with the local reversibility conditionof the bijectivization leaves us with the right-hand side of the desired identity (3.9).
Results of proposition 3.1.10 and proposition 3.1.11 extend naturally to the case when
we consider Yang-Baxter equations related to sqW polynomials. In the following we reportthese results omitting the details of proofs.
Proposition 3.1.12. Let pfwd and pbwd be a stochastic bijectivization of Yang–Baxter
equations (B.11), (B.26) for the weights W, w∗ , and Ufwd be constructed from sequential
local moves pfwd . Then the random field generated by Ufwd is a sqW/sHL field.
Proposition 3.1.13. Let pfwd and pbwd be a stochastic bijectivization of Yang–Baxter
equations (B.12), (B.26) for the weights W, W ∗ , and Ufwd be constructed from sequential
local moves pfwd . Then the random field generated by Ufwd is a sqW/sqW field.

3.1.7

Markov marginals of a Yang-Baxter field

By the very construction, we see that for any fixed c ≥ 1, the update (κ1 , . . . , κc ) →(ν1 , . . . , νc ) in a Yang-Baxter field is independent of κi , µi , λi for all i ≥ c + 1. Therefore,
we have:
Proposition 3.1.14. Let {λ(i,j) : i, j ∈ Zj∈≥0 } be a Yang–Baxter random field as above.
(i,j)
(i,j)
For any c ∈ Z≥1 , the marginal process {(λ1 ≥ · · · ≥ λc ) : i, j ∈ Z≥0 } is a Markov
process.
Proof. This is [BP19, Proposition 6.2].
In the simplest case c = 1, transition probabilities of the one-dimensional marginal
field can be written down explicitly:71Proposition 3.1.15. Let {λ(i,j) : i, j ∈ Z≥ 0} be a random field generated by Ufwd con(i,j)
structed from bijectivization of the Yang–Baxter equation. Let {λ1 : i, j ∈ Z≥0 } be thefirst row marginal process. Then for all i, j ≥ 1 we have(i,j)(i,j−1)= n | λ1(i−1,j)= m, λ1(i−1,j−1)= k} = Lui ,vi (m−k, `−k; n−`, n−m),(3.10)
for all n, m, k, n ≥ 0, where L is the stochastic vertex weightProb{λ1= `, λ1

Lu,v (j2 , j1 ; k10 , k20 ) = P

wr{0,0,∞},{j1 ,j2 ,∞} ({k10 , k20 , ∞}).
k1 ,k2 wl{0,0,∞},{j1 ,j2 ,∞} ({k1 , k2 , ∞})(3.11)

Note that interlacing implies that k ≤ m ≤ n, k ≤ ` ≤ n, so the arguments of Lui ,viin (3.10) are all nonnegative.
Proof of Proposition 3.1.15. This is proven in [BMP19, Section 6.4] and we briefly repro(i−1,j−1)(i,j)(i−1,j)(i,j−1)duce the argument here. The update λ1→ λ1 , once λ1, λ1
are fixed, isdetermined only by a single random move at the leftmost column of vertices. By construction, the vertical direction at the leftmost column has infinitely many paths. The
corresponding Yang–Baxter equation isXX
wl({k1 , k2 , ∞} | {0, 0, ∞}, {j1 , j2 , ∞}) =wr({k10 , k20 , ∞} | {0, 0, ∞}, {j1 , j2 , ∞}).
k10 ,k20

k1 ,k2(3.12)

This implies that taking
0
0
0
0
pfwd
{0,0,∞},{j1 ,j2 ,∞} ({k1 , k2 , ∞} → {k1 , k2 , ∞}) = Lu,v (j2 , j1 ; k2 , k1 )indeed produces a bijectivization.2 Here u, v denote generic spectral parameters of weights
appearing in the Yang–Baxter equation. Recall that occupation numbers are related tosignatures as(i−1,j)(i−1,j−1)(i,j−1)(i−1,j−1)
j1 = λ1− λ1,
j2 = λ1− λ1,(3.13)(i,j)(i−1,j)(i,j)(i,j−1).
,
k20 = λ1 − λ1k10 = λ1 − λ1This completes the proof.
The fact that the sqW functions are parametrized by signatures with specified numberof rows also allows to access the random dynamics of last rows of a field by writing down(i,j)
explicit bijectivizations. In particular, the evolution of {λi : i, j ≥ 0} is related to theYang–Baxter equations (B.26), (B.27) corresponding to configurations depicted in Figure
B.5 (b).
Remark 3.1.16. The construction of a random field using stochastic bijectivizations does
not guarantee that the evolution of last rows is autonomous. This contrasts with the factthat the first rows form autonomous Markov marginal processes by the very constructionof Yang–Baxter fields (Proposition 3.1.14). In Theorems 3.3.10 and 3.3.16 below we show(i,j)
that the marginals {λi : i, j ≥ 0} of sqW/sHL and sqW/sqW fields, respectively, are infact autonomous for a particular bijectivization we construct.
2This bijectivization is in fact unique for our choices of weights (this follows similarly to [BMP19]).
However, we do not need this fact.

72

3.2

Schur case: Robinson–Schensted–Knuth from Yang–
BaxterIn this section, as a simpler illustration, we consider the degeneration of the vertex weights
and the Yang–Baxter equation obtained by setting q = s = 0, and show how this produces (via bijectivization) the classical Robinson–Schensted–Knuth (RSK) row insertion
algorithm [Knu70], [Ful97], [Sta01]. We would obtain a “local” description of the RSK
insertion in terms of “toggle” operations. We refer to, e.g., [Pak01], [KB95], [Fom95] andalso to the recent notes [Hop14] for this description.
We consider the q = s = 0 degeneration of the Yang–Baxter equations ((B.12)(B.26))
proving the sqW/sqW skew Cauchy identity (Theorem 2.3.15). Note that for q = s =0, the spin q-Whittaker polynomials become the Schur polynomials. The Yang–Baxter
equations we need are illustrated in Figure B.5. In fact, in the Schur degeneration thecorner Yang–Baxter equation Equation (B.27) illustrated in Figure B.7 (b) is the same as
the usual one, and so we only need the equation from Equation (B.12).
The weights entering the Yang–Baxter equation degenerate as follows:
Wx,0 (i1 , j1 ; i2 , j2 ) = 1i1 +j1 =i2 +j2 1i1 ≥j2 xj2 ,∗Wy,0(i1 , j1 ; i2 , j2 ) = 1i1 +j2 =i2 +j1 1i2 ≥j2 y j2 ,Rx,y,s (i1 , j1 ; i2 , j2 ) = 1i2 +j1 =i1 +j2 (xy)min(j1 ,j2 )(3.14).

Equation (B.12) thus reads for all fixed i1 , i2 , i3 , j1 , j2 , j3 ∈ Z≥0 :X

k1 ,k2 ,k3 ≥0

1arrow preservation 1k3 ≥max(j1 ,j2 ) xj2 y j1 (xy)min(k1 ,k2 )=X

k10 ,k20 ,k30 ≥0001arrow preservation 1i3 ≥k20 1j3 ≥k10 (xy)min(j1 ,j2 ) xk2 y k1 , (3.15)

where by “arrow preservation” we mean the intersection of all the conditions of the form
a1 + b1 = a2 + b2 in (3.14) which the indices k1 , k2 , k3 and k10 , k20 , k30 must satisfy. In
particular, by arrow preservation we have
k2 = i2 + k1 − i1 ,k20 = j2 + k10 − j1 ,k3 = i3 + j1 − k1 ,k30 = i1 + j3 − k10 ,and thus the summands in the left- and right-hand sides of (3.15) are indexed only by k1
or k10 , respectively. Equation (3.15) admits a bijective proof (or, in terms of Section 3.1,
a bijectivization which is deterministic):
Lemma 3.2.1. Setting
k10 = j1 − min(j1 , j2 ) + min(k1 , i2 + k1 − i1 )produces a bijection between the terms in both sides of (3.15).

73(3.16)

Proof. By (3.16) we see that the powers of y in the corresponding terms match. The
powers of x match, too:
j2 + min(k1 , i2 + k1 − i1 ) = min(j1 , j2 ) + j2 + k10 − j1 ,where k10 is given by (3.16). It remains to check that if k1 is such that the product ofindicators in the left-hand side of (3.15) is nonzero, then the same holds for k10 in theright-hand side. This check is straightforward.
Let us now interpret a sequence of bijective Yang–Baxter transformations as a row
RSK insertion. Fix signatures λ, µ and use Lemma 3.2.1 to construct a bijection between
the sets{κ : κ ≺ λ, κ ≺ µ} × Z≥0 ←→ {ν : ν  λ, ν  µ} .
(3.17)
This bijection would correspond to a local move in the “Fomin growth diagram” [Fom95]
interpretation of the RSK.
Interpret µ, κ, λ as a path configuration as in Figure 3.2. The numbers of paths
through vertical edges are then equal to µc − µc+1 , κc − κc+1 , and λc − λc+1 . One can also
check that the horizontal edges carry µc − κc and λc − κc paths.
Attach a cross at the leftmost boundary of the path configuration. This cross is notuniquely determined since at the leftmost boundary (with i3 = j3 = ∞, i1 = i2 = 0) theYang–Baxter equation (3.15) takes the form
∞
X
X
0
0
x y
(xy)k = (xy)min(j1 ,j2 ) xj2 −j1
1arrow preservation xk1 y k1 .
j2 j1

k10 ≥0k=0

Arrow preservation in the right-hand side here means that there exists an arrow configuration with the given k10 .
Selecting an arbitrary k in the left-hand side is equivalent to selecting an element ofZ≥0 in the left-hand side of (3.17). Then we set based on the Yang–Baxter equation (seeLemma 3.2.1):
k10 = j1 − min(j1 , j2 ) + k.
In terms of signatures, k10 corresponds to the new number of paths on the horizontal edge,
and so the above equation means that
ν1 − µ1 = λ1 − κ1 − min(λ1 − κ1 , µ1 − κ1 ) + k,that is,ν1 = k + max(λ1 , µ1 ).

(3.18)

After dealing with the leftmost boundary, we move the cross one by one to the right,
updating each κc − κc+1 to νc − νc+1 , where c ≥ 2. At each step the signatures correspondto the path numbers as
j1 = λc+1 − κc+1 ,k10 = νc+1 − µc+1 ,j2 = µc+1 − κc+1 ,k1 = λc − κc ,00k2 = j2 + k1 − j1 = νc+1 − λc+1 .
74

k2 = i2 + k1 − i1 = µc − κc ,

The local bijection of Lemma 3.2.1 then leads to
k10 = j1 − min(j1 , j2 ) + min(k1 , k2 )= λc+1 − κc+1 − min(λc+1 − κc+1 , µc+1 − κc+1 ) + min(λc − κc , µc − κc )= λc+1 − min(λc+1 , µc+1 ) − κc + min(λc , µc ),which leads toνc+1 = max (λc+1 , µc+1 ) + min(λc , µc ) − κc .

(3.19)

Formulas (3.18)–(3.19) for ν in terms of κ provide the local RSK bijection between
the two sets (3.17). Moreover, these formulas have the “toggle” form, e.g., see [Hop14].
Therefore, we see that in the Schur degeneration the Yang–Baxter equation of Equation (B.12) produces a bijection, and this bijection coincides with the “toggle” bijectionin the local description of the classical Robinson–Schensted–Knuth row insertion.

3.3

Marginals of spin Hall-Littlewood and spin qWhittaker fields

In this section we characterize marginals of random fields defined in Section 3.1.2 corre(i,j)(i,j)
sponding to the first and last coordinates λ1 and λi . These are matched with stochastic vertex models or particle dynamics introduced in [GS92] [Pov13], [CP16], [CMP19].
In case of fields weighter by sqW functions, these results extend the characterization of
marginals of the q-Whittaker processes given in [MP17] by adding the spin parameter s
into the picture. The matchings are summarized in the table in Figure 3.3.
(i,j)

sHL/sHL fieldsqW/sHL fieldsqW/sqW field

first row λ1[3.3.2] Stochastic six vertex
model [GS92],[BCG16][3.3.4] Stochastic higher spin
six vertex model [CP16],[BP18a][3.3.6] 4 φ3 vertex model and
q-Hahn PushTASEP [CMP19],[BMP19](i,j)

last row λi[3.3.3] Stochastic higher spin six
vertex model [CP16], [BP18a][3.3.5, 3.3.7] q-Hahn TASEP /Boson particle systems [Pov13],[Cor14]

Figure 3.3: A summary of matchings of Section 3.3, with numbers of relevant subsections.

3.3.1

Stochastic vertex models

We work with two typologies of stochastic vertex models: up-right or up-left. Theseare probability measures on directed path ensembles (of the corresponding direction) in75

the integer quadrant, constructed from families of stochastic vertex weights L(i,j) . By
“stochastic” we mean that the weights must satisfy the sum to one conditionX
L(i,j) (α1 , β1 ; α2 , β2 ) = 1(3.20)
α2 ,β2 ≥0

for all α1 , β1 , where α1 , α2 , β1 , β2 ∈ Z≥0 are the occupation numbers of edges at a vertex(i, j).
For the first type of stochastic vertex models, equip the lattice with up-right vertex
weights Lur(i,j) subject to the arrow preservation condition
Lur(i,j) (α1 , β1 ; α2 , β2 ) = 0

if

α1 + β1 6= α2 + β2 .

Definition 3.3.1 (Up-right stochastic vertex model). The up-right stochastic vertex model
v
v v
h
h h
with weights Lur(i,j) and boundary conditions B = {b1 , b2 , . . . }, B = {b1 , b2 , . . . }, withbhi , bvj ≥ 0, is the unique probability measure on the set of up-right directed paths on
Z≥1 × Z≥0 , such that:• each vertex (1, j) emanates bvj paths initially directed to the right;• each vertex (i, 0) emanates bhi paths initially directed upwards;• the probability of observing a configuration (α1 , β1 ; α2 , β2 ) at vertex (i, j), conditioned on the configuration at all vertices (i0 , j 0 ) with i0 + j 0 < i + j, is given by
Lur(i,j) (α1 , β1 ; α2 , β2 ). Moreover, this event is independent of choosing arrow configurations at other vertices . . . , (i − 1, j + 1), (i + 1, j − 1), . . . on the same diagonal.
Up-right directed lattice path configurations can be encoded by the height function:
Hur (i, j) = #{occupations at horizontal edges} − #{occupations at vertical edges},(3.21)
1 1
1 1
11
where occupations are counted along the path ( 2 , 2 ) → (i + 2 , 2 ) → (i + 2 , j + 2 ) (equivalently, along any up-right directed path from ( 12 , 12 ) to (i + 12 , j + 21 )). See Figure 3.4,
right, for an illustration of the vertex model and the corresponding height function.
Remark 3.3.2 (Up-right model and TASEPs). Path configurations can be interpreted as
trajectories of particles performing totally asymmetric random walks, with time runningin the upward direction. In particular, one can define a process{X(t) = (x1 (t) > x2 (t) > · · · )}t∈Z≥0by setting xn (t) := Hur (n, t) − n. Then X is a discrete time totally asymmetric simple
exclusion process, in which the random jump xn (t − 1) → xn (t) of the n-th particle attime t is governed byLur(n,t) xn−1 (t − 1) − xn (t − 1) − 1, xn−1 (t) − xn−1 (t − 1); xn−1 (t) − xn (t) − 1, xn (t) − xn (t − 1) .
76

j
.
.
.
2
1
0

1

2

3

4

6

1

1

3

4

5

1

1

3

4

5

0

1

2

3

3

0

0

2

2

3

0

1

2

···

j
.
.
.
2
1
0

i

3

2

2

0

0

2

2

2

0

0

2

1

1

0

0

1

0

0

0

−1

0

0

−1 −1 −1

0

1

2

···

i

Figure 3.4: Realizations of the up-left and the up-right stochastic vertex models (left andright panels, respectively).
Let us now turn to up-left path ensembles. The up-left weights Lul(i,j) satisfy thefollowing arrow preservation property:
Lul(i,j) (α1 , β1 ; α2 , β2 ) = 0

if

α1 + β2 6= β1 + α2 .

Definition 3.3.3 (Up-left stochastic vertex model). The up-left stochastic vertex model
h
h h
v
v v
with weights Lul(i,j) and boundary conditions B = {b1 , b2 , . . . }, B = {b1 , b2 , . . . }, withbhi , bvj ≥ 0, is the unique probability measure on the set of up-left directed path on Z≥1 ×
Z≥0 , such that:• each vertex (1, j) has bvj paths entering from its left;• each vertex (i, 0) emanates bhi paths initially directed upwards;• the probability of observing a configuration (α1 , β1 ; α2 , β2 ) at a vertex (i, j), conditioned on the path configuration at vertices (i0 , j 0 ) with i0 + j 0 < i + j, is given by
Lul(i,j) (α1 , β1 ; α2 , β2 ). Moreover, this event is independent of choosing arrow configurations at other vertices . . . , (i − 1, j + 1), (i + 1, j − 1), . . . on the same diagonal.
Up-left directed lattice path configurations can be encoded by the height function:
Hul (i, j) = #{occupations at horizontal edges} + #{occupations at vertical edges},(3.22)
where occupations are counted along the path ( 21 , 12 ) → (i + 21 , 12 ) → (i + 21 , j + 12 ) (equivalently, along any up-right directed path from ( 21 , 12 ) to (i + 12 , j + 12 )). Notice the differencein sign with the definition of Hur (3.21). See Figure 3.4, left, for an illustration of the
up-left vertex model and the corresponding height function.
Remark 3.3.4 (Up-left model and PushTASEPs). Define a process{Y(t) = (y1 (t) > y2 (t) > · · · )}t∈Z≥077

by setting yn (t) = −Hul (n, t) − n. Then Y is a discrete time totally asymmetric simple
exclusion process under which particles jump to the left, and a pushing mechanism ispresent. The random jump yn (t − 1) → yn (t) of the n-th particle at time t is governed by

y
(t
−
1)
−
y
(t
−
1)
−
1,
y
(t
−
1)
−
y
(t);
y
(t)
−
y
(t)
−
1,
y
(t
−
1)
−
y
(t)
.
Lul
n−1
n
n−1
n−1
n−1
n
n
n
(n,t)
In the rest of this section we establish the matching results outlined in Figure 3.3.

3.3.2

First row in the sHL/sHL field(i,j)

Here we study the scalar Markov marginal λ1
match it to the stochastic six vertex model.

of the sHL/sHL Yang-Baxter field, and

i2
j1

j2
i1

Luv (i1 , j1 ; i2 , j2 )1uv(1−t)
1−tuv

1−uv
1−tuv

t(1−uv)
1−tuv

1−t
1−tuv

1

Figure 3.5: The vertex weights Luv (i1 , j1 ; i2 , j2 ) in the stochastic six vertex model. Thisparametrization of the vertex weights follows, e.g., [BM18].
Definition 3.3.5 ([GS92], [BCG16]). Specialize the up-right stochastic vertex model ofDefinition 3.3.1 by taking Lur(i,j) = Lui vj given in Figure 3.5. We refer to this model as thestochastic six vertex model. We consider the step boundary conditions:bvj = 1andbhi = 0,(3.23)The next theorem is analogous to [BP19, Proposition 7.3].
urTheorem 3.3.6. Let H6V(i, j) be the height function of the six vertex model with stepboundary conditions and let {λ(i,j) } be the sHL/sHL field. Then the two random fields(i,j)
ur{j − λ1 : i, j ∈ Z≥0 } and {H6V(i, j) : i, j ∈ Z≥0 } are equal in distribution.

Proof. Recall that, by proposition 3.1.15, the Markov evolution of the scalar marginal(x,y)λ1
is determined by the quantity L of eq. (3.11) corresponding to Yang-Baxter equation (B.2). Its expression can be readily written down using (2.2), (2.9), and Figure B.3. Comparing these quantities to the stochastic six vertex weights Luv in Figure 3.5,
(i,j)
while taking into account the relation between j2 , j1 ; k10 , k20 and λ1 (3.13), we see that
LsHL(u),sHL(v) (j2 , 1 − j1 ; k10 , 1 − k20 ) = Luv (j2 , j1 ; k10 , k20 ) for all j2 , j1 ; k10 , k20 ∈ {0, 1}. Thisimplies the claim.
Remark 3.3.7. While Theorem 3.3.6 essentially follows from [BP19], let us emphasize what is different here. Theorem 3.3.6 connects the ordinary stochastic six vertex
model with stable spin Hall-Littlewood polynomials. On the other hand, previously thesame stochastic six vertex model was matched to Hall-Littlewood processes and measures[BBW18], [BM18], and the non-stable spin Hall-Littlewood polynomials gave rise to a
dynamic version of the stochastic six vertex model [BP19]. Therefore, formally Theorem 3.3.6 is a new statement.
78

α2

g

g−1

g+1

g

α1

g

g

g

g

Lur
x,v (α1 , β1 ; α2 , β2 )1+q g xv
1+xv

xv(1−q g )1+xv

1+q g sx
1+xv

xv−q g sx
1+xv

β1

β2

Figure 3.6: The stochastic vertex weights Lur
x,v for the up-right stochastic higher spin six
vertex model.

3.3.3

Last row in sqW/sHL field

We start by defining the stochastic higher spin six vertex model:
Definition 3.3.8 ([CP16], [BP18a]). Specialize the up-right stochastic vertex model ofurDefinition 3.3.1 by taking Lur(i,j) = Lxi ,vj , where the latter are given in Figure 3.6. We
refer to this model as the up-right stochastic higher spin six vertex model. We consider
the step-stationary boundary conditions:


x1 vj
v
bj ∼ Berandbhi = 0,(3.24)
1 + x1 v j
where Ber(·) are independent Bernoulli random variables with the probability of successgiven in the parentheses.3
Remark 3.3.9. The model in Definition 3.3.8 is equivalent to that of [BP18a] (the latterwith step boundary conditions bvj = 1, bhi = 0), under specializations ξ1 s1 → x1 , s21 → 0,sα ξα → xα , s2α → −sxα and uβ → −vβ .
(i,j)

Theorem 3.3.10 (sqW/sHL last row). The last row marginal {λi }i≥1,j≥0 of the sqW/sHL
urfield has the same distribution as the height function {HHS(i, j)}i≥1,j≥0 of the up-righthigher spin six vertex model with step-stationary boundary conditions.
Proof. We use Proposition 3.1.12. During the update
λ(n−1,t−1) → λ(n,t) ,for fixedλ(n−1,t) , λ(n,t−1) ,(3.25)

weighted by the stochastic matrix Ufwd , the law of the rightmost local move is given by astochastic bijectivization of the Yang–Baxter equation (B.26). A computation shows thatone such bijectivization is given by the choice
ur
0
0
0
0
0
pfwd
{i1 ,i2 ,i3 },{∅,j2 ,j3 } ({k1 , k2 , k3 } → {k1 , k2 , k3 }) = Lx,v (j3 − i2 − k1 , k1 ; j3 − i2 − k1 , k1 ). (3.26)
3A slightly broader class of boundary conditions than the step-stationary ones, where also bhi areallowed to be positive numbers, can be considered using the fusion argument introduced in [Agg18]; seealso [IMS20], [BMP19].

79

This can be readily verified using the parametrization from Example 3.1.8. In terms ofelements of signatures in (3.25), the integers i2 , j3 , k1 , k10 are interpreted as(n,t−1)i2 = λn−1(n−1,t)k1 = λn−1(n−1,t)− λn−1(n,t−1),

j3 = λn−1(n−1,t−1)− λn−1,− λ(n,t−1)nk10 = λ(n,t)− λ(n,t−1).
nnRemarkably, transition weight (3.26) only depends on the difference j3 − i2 and on k1 , k10 ,(n,t)
but not on other edge occupation numbers. Therefore, the law of λn is solely determined(n−1,t) (n,t−1) (n−1,t−1)(i,j)
by λn−1 ,λn,λn−1. This implies that the last row marginal {λi }i≥1,j≥0 is anautonomous Markov process. Moreover, this autonomous process has the same multitime
joint distribution as the height function of the up-right higher spin six vertex model
because Lur appears in (3.26). This completes the proof.
In [CP16], [BP18a], joint q-moments of the up-right stochastic higher spin six vertex
model were expressed in terms of nested contour integrals. These moments completelyurdetermine the joint distribution of the model’s height function HHS(·, j) along any given
horizontal line (because q ∈ (0, 1) and the random variables in question are nonnegative).
Let us reproduce the q-moment formula:
Proposition 3.3.11 ([BP18a]). Consider the up-right stochastic higher spin six vertex
model with step-stationary boundary conditions and assume vα 6= qvβ . For any i1 ≥ . . . ≥i` ≥ 1 we have
E

Ỳ

k=1

q

ur (i ,j)HHSk`= q (2)Idz1···
γ[−v|1] 2πi

×

Ỳ

k=1

I

zA − zBdz` Y
γ[−v|`] 2πi 1≤A<B≤` zA − qzB

j
ik
Y
1
xα (1 + szk ) Y 1 + qvα zkzk (1 + szk ) α=1 xα − zk α=1 1 + vα zk!

(3.27).

Here, integration contours are γ[−v|k] = γ[−v]∪rk−1 c0 , where γ[−v] encircles −1/v1 , . . . , −1/vj
and no other singularity, c0 is a small circle around 0, and r > q −1 . All curves are positively oriented, and rk−1 c0 never intersects γ[−v] for k = 1, . . . , `.
Proof. This follows from Theorem 9.8 in [BP18a] by identifying the parameters as inurRemark 3.3.9 and noting that HHS(i, j) is the same as the height function h(i) at thej-th horizontal slice. Note also that [BP18a, Corollary 10.3] is essentially the same as
our q-moments (3.27), but with contours dragged through infinity, and identification ofs21 with x1 . The latter follows by comparing (3.24) with [BP18a, Remark 6.14].
Eigenrelations for sqW polynomials given in Theorem 2.6.11 can be employed to provide an alternative proof of the moment formula (3.27).
Alternative proof of Proposition 3.3.11. We express q-moments of last rows of the sqW/sHL
process using the q-difference operators D1 (2.85) at several levels, following the argument
in [Bor+16, Proposition 4.4].
80

(i)

Denote by D1 the operator D1 acting on i variables x1 , . . . , xi . Then for any ` andany sequence 1 ≤ i1 ≤ · · · ≤ i` , we have
(i )
(i )
Ỳ λ(ik ,j)D1 1 · · · D1 ` Π(x1 , . . . , xN ; v1 , . . . , vj )
ik
E
,
q
=
Π(x
1 , . . . , xN ; v1 , . . . , vj )k=1

where N ≥ i` is arbitrary, andΠ(x1 , . . . , xN ; v1 , . . . , vj ) =

j 
Y
r=1

1
1 − svr(3.28)

N −1 Yj
N Y
(1 + vr xi )i=1 r=1

is the partition function in the right-hand side of the sqW/sHL Cauchy identity (2.27).
Equality (3.28) is a straightforward consequence of the Cauchy identities (2.25), (2.27),
eigenrelation (2.87), and the branching rules for the sqW functions.
Let us now express the right-hand side of (3.28) in terms of nested contour integrals.
ForjYh(z) =(1 + vr z),
r=1

we have

(i )

(i )

D 1 · · · D1 ` h(x1 ) · · · h(xN )r.h.s. (3.28) = 1.
h(x1 ) · · · h(xN )Moreover, for any meromorphic function eh we have

I Y
n 


xα (1 + sz) e
h(qz)
1
dz
(n) e
e
e
h(xα ),
D1 h(x1 ) · · · h(xn ) =e
2πi γhe α=1xα − zh(z) z(1 + sz)

h(qz)/eh(z). The latter poles may include
where the curve γeh encircles 0 and all poles of e
infinity, too. In other words, the integral over γeh is equal to the sum of minus residues of
the integrand at x1 , . . . , xn .
By iterating this integral representation, we can evaluate (3.28) and match the result(i,j)
ing expression with the q-moment formula (3.27). The equivalence of processes λi andurHHS(i, j) stated in Theorem 3.3.10 allows us to complete the proof.

3.3.4

First row in sqW/sHL fieldLet us define an up-left version of the stochastic higher spin six vertex model. Takeulan up-left model in the sense of Definition 3.3.3, with the weights Lul(i,j) = Lxi ,vj , given in
Figure 3.7. We take this model with the same step-stationary boundary conditions (3.24).
In fact, this model is essentially the same as the one from Definition 3.3.8:
Remark 3.3.12. When at most one path occupies each horizontal edge (as in our case),swapping the horizontal occupation numbers 0 ↔ 1 is a bijection between up-left and upurulright models. Their height functions are related as HHS(i, j) = j − HHS(i, j). Moreover,ulur
the weights Lx,v become the weights Lx,v from Figure 3.6 after this swapping of horizontal
occupations, and the inversion of the parameters (x, v) 7→ (x−1 , v −1 ).
81

α2

g

g+1

g−1

g

α1

g

g

g

g

Lul
x,v (α1 , β1 ; α2 , β2 )1−q g sv
1+xv

xv+q g sv
1+xv

1−q g
1+xv

xv+q g
1+xv

β1

β2

Figure 3.7: The stochastic vertex weights Lul
x,v for the up-left stochastic higher spin six
vertex model.
However, it is convenient to work with the up-right and the up-left models separately,
as in the sqW/sqW case they are genuinely different.
(i,j)

Theorem 3.3.13 (sqW/sHL first row). The first row marginal {λ1 }i≥1,j≥0 of the sqW/sHL
ulfield has the same distribution as the height function {HHS(i, j)}i≥1,j≥0 of the up-leftstochastic higher spin six vertex model with step-stationary boundary conditions.
Proof. We use Yang–Baxter fields similarly to the approach taken in [BMP19, Section
7.3]. Let us specialize the general notation of Proposition 3.1.15. We need to match thestochastic vertex weight L of (3.11) with Lul , and verify boundary conditions.
(i−1,j)(i,j−1)(i,j)(i−1,j−1)
is determined, λ1→ λ1 , conditioned on λ1
The random move λ1
by the bijectivization of the Yang–Baxter equation (B.11) for i > 1, j ≥ 1 and by thebijectivization of (B.26), if i = 1. We start with the first case, where in (3.11) we get(after canceling common factors)
v j1(−s/x; q)j2
xj 2,
1 − sv(q; q)j20v k1
0 (−s/x; q)k 0
0
0
0
0
2
wr{0,0,∞},{j1 ,j2 ,∞} ({k1 , k2 , ∞}) = Rx,v,s (k2 , k1 ; j2 , j1 )xk2.
1 − sv(q; q)k20

wl{0,0,∞},{j1 ,j2 ,∞} ({k1 , k2 , ∞}) = Rx,v,s (0, 0; k1 , k2 )One readily sees that then (3.11) gives the stochastic weight Lul
x,v .
(1,j)
For the boundary signature λ1 case, configuration weights wl, wr become (after canceling common factors)
wl{0,0,∞},{∅,j2 ,∞} ({k1 , k2 , ∞}) = Rx,v,s (0, 0; k1 , k2 ) xj2 ,wr{0,0,∞},{∅,j2 ,∞} ({k10 , k20 , ∞}) =

0

v k1
0
xk 2 ,
1 − sv

which leads to the step-stationary boundary conditions (3.24) since λ(i,0) = ∅ for all i.

3.3.5

Last row in sqW/sqW field

Define the up-right stochastic weight byLurx,y (α1 , β1 ; α2 , β2 ) := 1α1 +β1 =α2 +β2 ϕq,xy,−sx (β2 | α1 ),where ϕ is the q-beta-binomial distribution (A.16)–(A.17).
82(3.29)

Definition 3.3.14 ([Pov13]). The q-Hahn vertex model is the up-right stochastic vertex
urmodel, in the sense of Definition 3.3.1, with weights Luri,j = Lxi ,yj . We consider stepstationary boundary conditions:bvj ∼ ϕq,x1 yj ,−sx1 (• | ∞)andbhi = 0,(3.30)

where the random variables for bvj are independent. Denote the corresponding heighturfunction by Hq-Hahn.
Remark 3.3.15. The model of Definition 3.3.14 is equivalent to that of [BP18a, Section
6.6.2], where parameters have been specialized as s2α → −sxα and q Jα → −yα /s.
(i,j)

Theorem 3.3.16 (sqW/sqW last row). The last row marginal {λi }i≥1,j≥0 of the sqW/sqW
urfield has the same distribution as the height function {Hq-Hahn(i, j)}i≥1,j≥0 of the up-rightq-Hahn vertex model.
Proof. This follows from Theorem 3.3.10 which established an analogous result matching
the last row of the sqW/sHL field and the height function of the up-right higher spin six
vertex model. By fusion, the dual sHL functions turn into the dual sqW functions (cf.
[BW17]). Therefore, the sqW/sHL field under fusion turns into the sqW/sqW field.
On the other hand, the same fusion procedure turns the up-right higher spin six vertex
model into the up-right q-Hahn vertex model4 . This completes the proof.
In [BP18a, Corollary 10.4] the multi-point q-moments of the up-right q-Hahn vertex
model were expressed in terms of nested contour integrals:
Proposition 3.3.17 ([BP18a], Corollary 10.4). Assume minα |sxα | > q maxα |sxα |. For
any i1 ≥ . . . ≥ i` ≥ 1 we have
E

Ỳ

k=1

q

ur
Hq-Hahn(ik ,j)`= (−1) q (2)

`

I

dw1
···
γ1+ [−sx] 2πi

×

Ỳ

k=1

I

wA − wBdw` Yγ`+ [−sx] 2πi 1≤A<B≤` wA − qwB

j
ik
Y
Y
1
1 − wk
1 + wk yα /swk (1 − wk ) α=1 1 + wk /(sxα ) α=1 1 − wk!

.

(3.31)

Integration contours encircle −sx1 , −sx2 , . . . , and leave out 0,1 and are q-nested in the+[−sx] is inside γk+ [−sx] for all k = 1, . . . , ` − 1.
sense that qγk+1
Proposition 3.3.17 was obtained in [BP18a] as a corollary (under fusion) of the multipoint q-moment formula (3.27) for the up-right higher spin six vertex model. Both ofthese q-moment formulas have several different proofs: via duality [CP16], manipulations
with symmetric functions using Bethe Ansatz [BP18a], or distributional matchings anddifference operators [OP17]. Eigenrelations for the sqW polynomials provide yet anotherindependent proof:
4

For a practical explanation of fusion in the context of sl2 stochastic vertex models see [BW17] and[CP16], [BP18a] and references therein.

83

Alternative proof of Proposition 3.3.17. Similarly to the alternative proof of Proposition 3.3.11
given in Section 3.3.3, we will use eigenrelations of the sqW polynomials to compute qmoments. To express q-moments of the sqW/sqW field, we use formula (3.28), after
replacing the function Π with the right-hand side of (2.34). The action of the difference
operator D1 (2.85) (in n variables) on a meromorphic function eh can be written as

I
n 


Y
dz
xα (1 + sz) e
h(qz)
1
e
e
e
h(xα ),
D1 h(x1 ) · · · h(xn ) = −e
2πi x1 ,...,xn α=1xα − zh(z) z(1 + sz)

where the integration contour contains x1 , . . . , xn , but doesn’t contain 0 or any pole ofeh(qz)/eh(z). Using this formula repeatedly, we can match the q-moments of the marginal(i,j)
λi to expression (3.31). The equivalence of processes between last row of the sqW/sqW
field and height function of the q-Hahn vertex model stated in Theorem 3.3.16 yields theproof.

3.3.6

First row in sqW/sqW field

For our fourth and final vertex model, define the up-left stochastic weight by1

s x
q β1 β2 + 2 α1 (α1 −1) (−s/x; q)α2 (−s/y; q)β2(−s/x; q)α1 (−s/y; q)β1 (q; q)β2 (−q/(sy); q)β2 −α2(s2 q α1 +β2 ; q)∞ (xy; q)∞q −β1 ; q −β2 , −sx, −q/(sy)q, q ,
×4 φ3−s/y, q 1+α1 −β1 , −xq 1−β2 −α1 /s(−sy; q)∞ (−sx; q)∞(3.32)yLulx,y (α1 , β1 ; α2 , β2 ) := 1α1 +β2 =α2 +β1

α2 α1 α2 −α1where 4 φ3 is the regularized q-hypergeometric function (A.10).
Remark 3.3.18. An expression equivalent to (3.32) for the stochastic weight Lul
x,y is givenbymin(`,L)Lul
x,y (g, `; g + L − `, L) =X
k=0

ϕq−1 ,qg ,−syqg−1 (k | `) ψq,−qk s/y,−qg s/x,s2 qg+k (L − k), (3.33)

where we used the q-beta-binomial and the q-hypergeometric distributions (A.16), (A.20).
This can be proved through simple manipulations of the q-Pochhammer terms. From
(3.33) it is immediate to see that Lul
x,y possesses the sum to one property (3.20). The
positivity of the weights (under certain restrictions on the parameters) is proven in section B.3.
Definition 3.3.19. The 4 φ3 vertex model is the up-left stochastic vertex model, in theulsense of Definition 3.3.3, with weights Luri,j = Lxi ,yj . We consider the same step-stationary
boundary conditions as in (3.30). The height function of this model is denoted by Hφul .
√Theorem 3.3.20 (sqW/sqW first row). Let s ∈ (− q, 0). The first row marginal(i,j){λ1 }i,j∈Z≥0 of the sqW/sqW field has the same distribution as the height function {Hφul (i, j)}i,j∈Z≥0of the 4 φ3 stochastic vertex model.
84

Proof. The proof of this matching is similar to that of Theorem 3.3.13, and follows from
Proposition 3.1.15. Namely, we specialize formula (3.11) using the Yang–Baxter equations(i−1,j−1)(i,j)(B.12), (B.27). For updates of “bulk” transition λ1→ λ1 , for i > 1, j ≥ 1,(i,j−1)(i−1,j)conditioned on λ1, λ1, the stochastic weight (3.11) uses(−s/y; q)j1 j2 (−s/x; q)j2x,(q; q)j1(q; q)j20 (−s/y; q)k 00 (−s/x; q)k 0
1
2
xk2
.
wr{0,0,∞},{j1 ,j2 ,∞} ({k10 , k20 , ∞}) = Rx,y,s (k20 , k10 ; j2 , j1 ) y k1
(q; q)k10(q; q)k20

wl{0,0,∞},{j1 ,j2 ,∞} ({k1 , k2 , ∞}) = Rx,y,s (0, 0; k1 , k2 ) y j1

Using the expression of the R-matrix Rx,y,s and summation identity (B.10) one can match(1,j)
Lx,y with Lx,y . At the boundary λ1 , we use a stochastic bijectivization of (B.27) andtherefore in this case we havewl{0,0,∞},{∅,j2 ,∞} ({k1 , k2 , ∞}) = Rx,y,s (0, 0; k1 , k2 ) xj2 ,0 (−s/y; q)k 0 (−sy; q)∞01
wr{0,0,∞},{∅,j2 ,∞} ({k10 , k20 , ∞}) = y k1
xk2 ,2
(q; q)k10(s ; q)∞that yields boundary conditions (3.30) after using again summation identity (B.10).

3.3.7

Push–block dynamics for sqW/sqW process

Let us now present another, more explicit matching of last rows of the sqW/sqW field
in a “Plancherel” (or “Poisson-type”) continuous time limit. Here the dynamics of thelast rows is matched to the corresponding continuous time limit of the q-Hahn TASEP.
This construction is very similar to how the continuous time q-TASEP emerges from
q-Whittaker processes in [BC14].
Consider the Borodin–Ferrari forward transition map (cf. Section 3.1.4)
Ufwd
x,y (κ → ν | λ, µ) =Fν/λ (x)F∗ν/µ (y)P,Π(x; y) κ Fµ/κ (x)F∗λ/κ (y)(3.34)∞ (−sy;q)∞where Π(x; y) = (−sx;q). In the limit as y = −s + ε(1 − q), ε → 0, the dual sqW(xy;q)∞ (s2 ;q)∞function at a single variable becomes (we use the notation [r]q = (1 − q r )/(1 − q))
λ = µ;
1 + O(ε),
∗
r−1
µ
−λ
+1
i−1
i
Fλ/µ (−s+ε(1−q)) =(−s) (q; q)rε+ O(ε2 ), λ = µ + rei for some i, r > 0.
µ−λ2[r]q (q i−1 i s ; q)rsee (2.31). Take yj = −s + ε(1 − q) for all j and rescale M = bt/εc, t ∈ R≥0 , inthe sqW/sqW field. Thus, we get a continuous time dynamics on interlacing arrays
λ1 (t) ≺ λ2 (t) ≺ · · · , where at time t, each λki jumps to λki + r, r ≥ 1, according to an
exponential clock with rate (see (3.34))kkrate(λ → λ + rei | λ

k−1

k−1
k−1
k
k
k
k
r−1
(−q λi −λi s/xk , q λi −λi+1 +1 , q λi−1 −λi +1−r ; q)rr (−s).
) = xkk−1+1 , q λki −λki+1 s2 , −q λi−1 −λki −r sx ; q)[r]q (q λki −λk−1ikr(3.35)

85

ej = λj + rei , the signature λj+1 isWhen an update occurs at level j bringing λj → λej+1 in the following way:
instantaneously updated to λej ≤ λj+1 , then λej+1 = λj+1
• if λiiej > λj+1 , then assume λej − λj+1 = m and set λej+1 = λj+1 + (m + `)ei with• if λiiii
probability
prob(λj+1

ej+1 | λj → λej ) = lim P→λε→0Fλej+1 /λej (x)F∗λej+1 /λj+1 (y)∗ej (x)Fη/λj+1 (y) y=−s+ε(1−q)η=λj+1 +(m+`0 )ei Fη/λ.

for any ` ≥ 0 (for ` large enough this probability vanishes). See Figure 3.8 for an
illustration.

ej+1 |λj → λej )prob(λj+1 → λ

ek |λk−1 )rate(λk → λ

λki+1

ek
λ
i

λki

λk−1
i

λk−1
i

j+1

k

λj+1
i

ej+1 λj+1
λ
i−1
i
j

k−1

λji

(a)

ej
λ
i

(b)

Figure 3.8: Push–block mechanism in the half-continuous sqW/sqW field. Each λki jumps
ek = λk + r at rate (3.35), which only depends on λk−1 ; see left panel. When a jump
to λii
happens at level k and breaks interlacing, it triggers an instantaneous push at levels above
to re-establish interlacing; see right panel.
When s = 0 and q ∈ (0, 1) in our dynamics, we recover the continuous time qWhittaker 2d-growth model introduced in [BC14, Definition 3.3.3]. Further setting q = 0
brings the original Borodin–Ferrari’s push–block process corresponding to Schur measures[BF14]. Note that in our case, in contrast with the Schur and q-Whittaker situations,jumps are long range.
Restricting attention to the last rows (leftmost diagonal) of the array and setting
k−1. Moreover, the pushingi = k in (3.35), we see that the rate only depends on λkk and λk−1
mechanism does not affect the leftmost diagonal of the array. Thus, the marginal evolutionof the particles in the leftmost diagonal is an autonomous Markov process. Its jump rates
are
k−1
k
r−1
(q λk−1 −λk +1−r ; q)rk−1
kkr (−s)
rate(λk → λk + r | λk−1 ) = xk.
k−1
−λkk −r[r]q (−q λk−1
sxk ; q)r
These rates correspond to an inhomogeneous version of the continuous time q-Hahn
TASEP studied in [BC16b], which is also a continuous time degeneration of the q-Hahn
86

TASEP of [Cor14]. Thus, we see that the continuous time push–block dynamics in thesqW case agrees with the last row marginal evolution.

3.4

Random boundary conditions

In this section we show how to generalize the construction of the Yang-Baxter fields given
in sections 3.1 and 3.3 in order to produce fields with random boundary conditions.

3.4.1

Fields with scaled geometric specializations

Stochastic bijectivizations of Yang-Baxter equations with scaled geometric specializations(B.22), (B.24) can be employed to produce fields with random boundary conditions. Fix
additional parameters α, β ∈ [0, −s−1 ], and consider specializations ρvi , ρhi , i = −1, 0, 1, . . .:
ρh−1 = sg(α),ρv−1 = sg(β),ρhi = sHL(ui ),ρvj = sHL(vj ),(3.36)

as in section 2.5. Let η be the Yang-Baxter field on the lattice Z≥−1 × Z≥−1 generated
by the Markov transition operators Ufwd
. Restricting the field η to the nonnegative
ρv ,ρhjiquadrant, denote λ = η|Z≥0 ×Z≥0 . We call λ the sHL/sHL Yang-Baxter field with two-sided
stationary boundary conditions.
Proposition 3.4.1. The single-point distributions in the sHL/sHL Yang-Baxter field withthe two-sided stationary boundary conditions are given by
Y 1 − ui0 vj 0(αβ; q)∞Prob{λ(i,j) = ν} = j
Fν 0 (u1 , . . . , ui ; ᾰ)F∗ν 0 (v1 , . . . , vj ; β̆).
i
0 vj 0QQ1−qui(1 + vj 0 α)
(1 + ui0 β) 1≤i00 ≤i
j 0 =11≤j ≤ji0 =1

Joint distributions in this field along down-right paths are expressed through products ofskew functions similarly to eq. (3.2).
Proposition 3.4.2. Consider positive specializations
ρhi = sqW(xi ),ρv−1 = sg(β),ρvj = sHL(vj ),i, j ≥ 0(3.37)and let η be the Yang-Baxter field on the lattice Z≥0 × Z≥−1 generated by the Markov
transition operators Ufwd
. Denote λ = η|Z≥0 ×Z≥0 . We call λ the sqW/sHL Yang-Baxter
ρv ,ρhjifield with two-sided stationary boundary conditions. Then λ is an sqW/sHL field with
mixed specializations. In particular its single-point distribution is given byQY(1 − svj 0 )i−11≤i0 ≤i (βxi0 ; q)∞(i,j)
QFν (x1 , . . . , xi )F•ν 0 (v1 , . . . , vj , β̆).
Prob{λ= ν} =
00(−sβ; q)i−1
(1
+
v
x
)
j i
∞
1≤i0 ≤i1≤j 0 ≤jRemark 3.4.3. The use of Yang-Baxter equations with corner weights automatically
produces random entries along the vertical boundary. This is the reason why to produce
a random sqW/sHL field we only employ scaled geometric specializations in the dual
term.
87

Proposition 3.4.4. Consider positive specializations
ρhi = sqW(xi ),ρv−1 = sg(β),ρvj = sqW(yj ),i, j ≥ 0(3.38)and let η be the Yang-Baxter field on the lattice Z≥0 × Z≥−1 generated by the Markov
transition operators Ufwd
. Denote λ = η|Z≥0 ×Z≥0 . We call λ the sqW/sqW Yang-Baxter
ρv ,ρhjifield with two-sided stationary boundary conditions. Then λ is an sqW/sqW field with
mixed specializations. In particular its single-point distribution is given byQY  (s2 ; q)∞ i−1 Y (xi0 yj 0 ; q)∞1≤i0 ≤i (βxi0 ; q)∞(i,j)
Prob{λ= ν} =Fν (x1 , . . . , xi )F∗ν 0 (y1 , . . . , yj
0 ; q)∞(−sβ; q)i−1
(−sy
;
q)
(−sx
j
∞
i
∞
0
1≤j 0 ≤j1≤i ≤i1≤j 0 ≤j

3.4.2

Vertex models with double sided stationary boundary conditions

Let us now turn to the marginal of fields with the two-sided stationary boundary condi(i,j)
tions described in Section 3.4.1. We will consider the behavior of λ1 at the boundary:
Proposition 3.4.5. Consider the transition probabilities L given in (3.11). Then we have0(vα)k100,
Lsg(α),sHL(v) (0, j1 ; k1 , k2 ) = 1j1 +k10 =k20

1 + vα0(uβ)k200
LsHL(u),sg(β) (j2 , 0; k1 , k2 ) = 1j2 +k20 =k10,
1 + uβ0(αβ)k2
Lsg(α),sg(β) (0, 0; k10 , k20 ) = 1k10 =k20
(αβ; q)∞ ,(q; q)k200 (−s/x; q)k 0 (βx; q)∞1
Lsg(β),sqW(x) (0, j1 ; k10 , k20 ) = 1j1 +k10 =k20 (βx)k1,(q; q)k10 (−sβ; q)∞0(βx)k200
Lsg(β),sqW (x) (0, 0; k1 , k2 ) = 1k10 =k20
(βx; q)∞ .
(q; q)k20(3.39)(3.40)(3.41)(3.42)(3.43)

The cases (3.39), (3.42) correspond to the bottom of the lattice and the left boundaries
and (3.41), (3.43) arise from the bottom left corner. Finally (3.40) is found at the verticalboundary of the two sided stationary sHL/sHL Yang-Baxter field. Observe that (3.41)
defines the q-Poisson distribution (cf. eq. (A.19)).
Proof of Proposition 3.4.5. All expressions follows by direct evaluation of (3.11) with theindicated choice of vertex weights. Normalization constant always come from the correct
specialization of (B.10) when evaluating the sum of the cross vertex weight.
Now take the stochastic six vertex model with independent Bernoulli boundary conditions (we call these the two-sided stationary boundary conditions:




ui β
1
h
v
bi ∼ Berandbj ∼ Ber.
(3.44)
1 + ui β
1 + vj α88That is, given a realization of these random variables, we then consider the stochastic six
vertex model with these boundary conditions according to Definition 3.3.5. We have thefollowing
Theorem 3.4.6. Let M be a q-Poisson random variable with parameter αβ independentof the stochastic six vertex model with two-sided stationary boundary conditions (3.44).
Let {λ(i,j) : i, j ≥ 0} be a sHL/sHL field two sided stationary boundary conditions. Then(i,j)
ur
two random fields {j − λ1 : i, j ∈ Z≥0 } and {H6V(i, j) − M : i, j ∈ Z≥0 } are equal in
distribution.
Proof. Theorem 3.4.6 follows in essentially the same way as Theorem 3.3.6 by matching
the value of vertex weights Luy ,vx and probability laws of entries bhx , bvx with those given by
L (on the boundary this follows from Proposition 3.4.5). We will not repeat the argument.
Result of theorem 3.4.6 extends with little modifications to all other marginal fields
we considered in section 3.3 and summarized in fig. 3.3. We report the cases concerningthe λ1 marginal in the following list of theorems.
Theorem 3.4.7. Let λ = {λ(i,j) : i, j ≥ 0} be a sqW/sHL field with mixed specializations(i,j)
and let M ∼ q-Poi(βx1 ) be independent of λ. Then the marginal field {λ1 : i, j ≥ 0} isululis the height function(i, j) + M; i, j ≥ 0}, where HHS
equivalent in distribution to {HHSof the up-left higher spin vertex model with boundary conditions

v j x1hv
bi ∼ ϕq,xi β,−s/xi (•|∞),
andbj ∼ Ber
.
(3.45)
1 + vj x1
Proof. The proof is equivalent to that of theorem 3.4.6 and consist in matching weights Lwith vertex weights and boundary conditions of the up-left higher spin vertex model.
Theorem 3.4.8. Let λ = {λ(i,j) : i, j ≥ 0} be a sqW/sqW field with mixed specializations(i,j)
and let M ∼ q-Poi(βx1 ) be independent of λ. Then the marginal field {λ1 : i, j ≥ 0} isulequivalent in distribution to {Hφul (i, j) + M; i, j ≥ 0}, where HHS
is the height function of
the 4 φ3 vertex model with boundary conditions
bhi ∼ ϕq,xi β,−s/xi (•|∞),

andbvj ∼ ϕq,x1 yj ,−sx1 (•|∞).

(3.46)

We conclude our discussion about marginals of fields with random boundary conditions
with the following remark.
Remark 3.4.9. It is possible to extend the characterization of marginals of fields with(i,j)
two-sided stationary boundary conditions to include the last row case λi . Here theresulting processes would be the up-right higher spin vertex model with boundary conditions (3.45) and the q-Hahn vertex model with boundary conditions (3.46) respectively
for the sqW/sHL field and the sqW/sqW field. To justify our claim we should construct
sqW functions labeled by signatures λ with integer parts (possibly negative) and define
yet another class of specializations for dual functions analogous to the scaled geometricones. The scheme is the one employed in [IS19]. We will not repeat the construction here.
89

3.5

Fredholm determinants from q-moment formulas

Here we derive Fredholm determinant expressions for the q-Laplace transform of the(i,j)
random variable λ1 , where λ = {λ(i,j) } is one of the Yang-Baxter fields described in
Section 3.1.2 In the sHL/sHL case, these Fredholm formulas are known [BCG16], [Agg18].
In the sHL/sqW case, they are present in the literature for the step and step-stationary
boundary conditions [CP16], [BP18a], [BP18b]. A Fredholm determinantal formula forthe stochastic higher spin six vertex model appears also in [IMS20] and we recall it below
in chapter 4, though the expression we establish here is different for this case. In thesqW/sqW case, a similar Fredholm formula for the q-Hahn PushTASEP was conjectured
in [CMP19], and here we prove this conjecture.
For the sake of this section in many cases we use the Borodin-Wheeler’s version of the(i,j)
sqW functions, as for the sake of marginal λ1 there is no difference between the two(i,j)
variants. The derivation of Fredholm determinantal formulas for marginals λi can be
carried on the same footing, but for brevity we will not treat this case here.

3.5.1

Six vertex model observables through difference operators

In this subsection we rederive known results about the q-moments of the six vertex model[BCG16], [BP18a] making use of the difference operators acting on spin Hall-Littlewood
functions. Consider the inhomogeneous stochastic six vertex model with the step boundur. Recall that the model depends on the parameters
ary conditions and height function H6V
vj and ui , i, j ∈ Z≥1 , which we assume positive (cf. definition 3.1.4). Let v1 , v2 , . . . bespaced in such a way thatq sup{vj } < inf {vj }.
(3.47)
jj

Proposition 3.5.1. Under (3.47) we haveII
ur (i,j) step l H6Vl(l−1)/2···Eq=qγ[v|1]YzA − zBγ[v|l] 1≤A<B≤l zA − qzB( j)i
Y qzk − vα Y
1 − zk uα dzk
×,zk − vα α=1 1 − qzk uα 2πizk
k=1 α=1lY(3.48)

where the positively orientated contour γ[v|j] = γv ∪ rj−1 C0 for zj is the union of a curve
γv that encircles v1 , . . . , vj and no other pole of the integrand, and the dilation rj−1 C0 of
an arbitrary small circle C0 around 0. Moreover, r > q −1 , and the shifted contour qγv
must lie completely to the left of γv and completely to the right of rl−1 C0 .
Proof. This is an application of the eigenrelations from Theorem 2.6.2. Define the operator
e as
D


∗
−j
e
:=
q
Id
+
(q
−
1)D
(3.49)
D
1 ,acting on v variables and diagonally on the sHL functions ase F∗λ (v1 , . . . , vj ) = q −`(λ) F∗λ (v1 , . . . , vj ).
D
90This operator, acting on product functions H(v1 , . . . , vj ) = h(v1 ) · · · h(vj ) admits an integral representation
1eDH(v
1 , . . . , vj ) = H(v1 , . . . , vj )
2πi

I

j
Y
w − q −1 vα

γ0,v α=1

w − vα

1 dw.
h(w) w(i,j)

ur
From theorem 3.3.6 we have the identification H6V(i, j) = j − λ1sHL/sHL field. Therefore, we havewhere

(3.50), where λ(i,j) is thee l Π(u1 , . . . , ui ; v1 , . . . , vj )Dur
Estep q l H6V (i,j) = q lj,
Π(u1 , . . . , ui ; v1 , . . . , vj )Π(u1 , . . . , ui ; v1 , . . . , vj ) =jiYY
1 − quβ vα

α=1 β=1

1 − uβ vα.

The nested contour formula (3.48) follows by recursively applying integral expression
e on factorized functions.
(3.50) for the action of D

Proposition 3.5.1 combined with well-known manipulations of summations of nested
contour integrals like (3.48) (e.g., see [BCS14, Section 3]) give rise to a Fredholm deterur.
minant5 expression for the one-point distribution of H6V
Theorem 3.5.2. Consider the stochastic six vertex model with step boundary conditions.
We have


1
step
E
= det (Id + K)L2 (C) ,ζ ∈ C \ R>0 .
(3.51)ur(ζq H6V (i,j) ; q)∞The expression in the right-hand side of (3.51) is the Fredholm determinant of the kernelZ(−ζ)r f(w)/f(q r w)
1
0
K(w, w ) =dr,(3.52)
2i d+iR sin(πr) q r w − w0
where d ∈ (0, 1), andf(w) =jY(w − vα )−1

α=1

2

i
Y

(1 − uα w).

α=1

The kernel K is defined on the Hilbert space L (C), where C is a closed positively oriented
curve encircling 0, v1 , v2 , . . . such that, for all r ∈ d + iR, C contains q r C but not q −r u−1i
for i = 1, 2, . . . .
We present the main steps of the proof of the Fredholm determinantal formula, andrefer to [BC14] or [BCS14] for detailed explanations.
5

On Fredholm determinants in general see, e.g., [Bor10].

91

Idea of proof of Theorem 3.5.2. Assume first (3.47) and |ζ| < 1/q, and consider the nested
contour expression (3.48). We can deform all contours, one by one, to be the same Caround 0, v1 , v2 , . . . , and such that C contains its image under multiplication by q. This
contour shift will cross poles zA = qzB , A < B, and one can rewrite (3.48) asX

1(q; q)l
m1 !m2 ! · · ·
λ`l

Z

C

···

Z

`(λ)

det

C i,j=1



1
w i − q λj w j

Y
`(λ)

f (wk )/f (q λk wk )

k=1

dzk
,
2πi

where the sum is taken over all partitions λ of l, and mi = mi (λ) are the multiplicities ofthe parts i in λ. Summing over l, we have


X ζl
1
ur (i,j) step
step l H6V=EqE,
ur (i,j)
H6V
(q;
q)
(ζq
;
q)
l
∞
l≥0
ur

where we used the absolute summability of the left-hand side (since 0 < q l H6V < 1 and
we assumed |ζ| < 1/q) to exchange the summation with the expectation sign and theq-binomial theorem. The result we obtain is the Fredholm determinant of the kernel
ZXζn1(−ζ)r f(w)/f(q r w)
nf(w)/f(qw)=.
0 − qnwr w − w0w
2i
sin(πr)qd+iR
n≥0Once we reach (3.51), we can relax conditions on ui ’s and ζ since both side are analytic
functions of their parameters. Formula (3.51) holds for any choice of ui , vj ∈ (0, 1) andζ ∈ C \ q Z≥0 (in particular, we can always find d in (3.52) such that C satisfies the required
properties).
In the next theorem we perform fusion of the sHL parameters. Recall the principal specializations F(J0 ,...,Ji ) (u0 , . . . , ui ), G(I0 ,...,Ij ) (v0 , . . . , vj ) defined in (2.38), (2.39). Parameters
ul , Jl , vk , Ik are complex numbers satysfying (2.40) which we reproduce here:|s|, |uk |, |vl |, |q Jk uk |, |q Il vl |,q i vl − s
q i uk − s,<δ
1 − q i suk
1 − q i svl

for all 0 ≤ k ≤ y, 0 ≤ l ≤ x, i ≥ 0.

(3.53)
for sufficiently small δ > 0 which might depend on x, y, but not on the other parameters.
Theorem 3.5.3. With the above notation, we have for all ζ ∈ C \ R>0 :Y (uα vβ ; q)∞ (uα vβ q Iβ +Jα ; q)∞(uα vβ q Iβ ; q)∞ (uα vβ q Jα ; q)∞
0≤α≤i

0≤β≤j

×

X F(J0 ,...,Ji ) (u0 , . . . , ui )G(I0 ,...,Ij ) (v0 , . . . , vj )λλλ(ζq −`(λ) ; q)∞92= det (Id + K)L2 (C) .
(3.54)

The kernel K is defined as
1
K(w, w ) =2i
0r−ζ f (w)/f (q r w)
dr,
q r w − w0
d+iR sin(πr)

Z

where d ∈ (0, 1) andjiY(q Iα vα /w; q)∞ Y (uα w; q)∞f (w) =.
(vα /w; q)∞ α=0 (q Jα uα w; q)∞α=0

The contour C is a closed positively oriented curve encircling 0, q k vi for k, i ≥ 0 and suchthat, for all r ∈ d + iR, C contains q r C and q r+k q Il vl for all k, l ≥ 0, but leaves outside
1/(q r+k ul ) and 1/(q k q Jl ul ) for all k, l ≥ 0.
Proof. Considering principal specializations in Theorem 3.5.2, we see that (3.54) holds
for any J0 , . . . Ji , I0 , . . . , Ij positive integers. Indeed, this follows from the computation
for I, J ∈ Z≥1 :q r w − vq I−1 1 − uw 1 − quw
1 − q J−1 uw
q r w − v q r w − qv
...
.
.
.
w − v w − qv
w − vq I−1 1 − q r uw 1 − q r uvw
1 − q r q J−1 uw(vq I /w; q)∞ (q −r v/w; q)∞ (uw; q)∞ (uq J wq r ; q)∞= q rI.
(v/w; q)∞ (q −r vq I /w; q)∞ (uq J w; q)∞ (uwq r ; q)∞(3.55)

The factor q rI (leading to q r(I0 +...+Ij ) in the kernel) disappears after replacing ζ byζq −I0 −...−Ij . This change of variable accounts for the fact that in the left-hand side of(3.54) we take the q-Laplace transform of q −`(λ) as opposed to the height function in
Theorem 3.5.2.
By the absolute convergence result of Proposition 2.4.4 and the boundedness of 1/(ζq −`(λ) ; q)∞ ,the left-hand side of (3.54) is an analytic function of q Jk , q Il under the bounds (3.53). In
order to establish the analyticity of the Fredholm determinant we first observe that, due
to the compactness of C and of the image of r → q r for r ∈ d + iR, there exists a constant
M1 independent of Jk or Il such thatf (w)/f (q r w)
< M1 .
q r w − w0
w,w0 ∈C,supr∈d+iR

This implies that |K(w, w0 )| < M2 integrating over r due to the exponential decay of
1/ sin πr for large |r|. We can thus estimate the Fredholm determinant of K withZ
X1Z
X ll/2 M ll3···det (K(wi , wj )) dw1 · · · dwl ≤
,
i,j=1
l!
l!
C
C
l≥0
l≥0
where we used the Hadamard inequality to bound the determinant of K(wi , wj ), andM3 = M2 `(C). This shows that the right-hand side of (3.54) is an absolutely convergent
sum of analytic functions and hence it is analytic. This completes the proof.
93

Remark 3.5.4. Fredholm determinantal expression (3.54) degenerates to a number ofknown results. In particular, considering the specialization v0 = −α, u0 = −β q I0 =q J0 = 1/,  → 0, and J1 = . . . = Ji = I1 = . . . = Ij = 1, we recover the expression
for the q-Laplace transform of the height function of the six vertex model with two-sided
stationary bound conditions from [Agg18, Proposition 4.1] (in the latter one has to setµ = 0). The latter formula is obtained by a more involved analytic continuation in q I0
than in the proof of Theorem 3.5.3.

3.5.2

Higher spin six vertex model observables

The eigenrelations for the sqW or sHL functions give rise to moment formulas for thestochastic higher spin six vertex model. Consider the model with step-Bernoulli boundary
conditions (see eq. (3.24)). Assume that the parameters x1 , x2 , . . . are spaced in such a
way thatq sup(1/xi )i≥1 < inf(1/xi )i≥1 .
(3.56)
e
Following the same approach as in the proof of Proposition 3.5.1 (applying either D
or D1 from Section 2.6 to the sum of the corresponding Cauchy identity), we obtain aq-moment formula which was essentially first written down in [CP16]:
Proposition 3.5.5. We haveIul (i,j)−l HHSl l(l−1)/2
E q= (−1) qΓ[x|1]

×

···

I

Y

zA − zBΓ[x|l] 1≤A<B≤l zA − qzB

( i
j
l
Y
Y 1 + szk Y
zk + vα /q

k=1

α=1

1 − xα zk α=1 zk + vαdzk
2πizk (1 + szk ))(3.57),

where the positively oriented contour Γ[x|k] is around 1/x1 , . . . 1/xi , qΓ[x, β|k + 1], andno other pole of the integrand.
ul
The following determinantal expression for the q-Laplace transform of the height HHS
is obtained specializing the general formula (3.54).

Theorem 3.5.6. Consider the higher spin six vertex model with two-sided stationary
boundary conditions (3.45) and let M ∼ q-Poi(x1 β) be independent of the vertex model.
Then we have1E
= det(Id + K)L2 (C) .
(3.58)ul(ζq −HHS (i,j)−M ; q)∞The kernel K is defined byZ1(−ζ)r f (w)/f (q r w)
0
K(w, w ) =dr,(3.59)
2i d+iR sin(πr) q r w − w0
where d ∈ (0, 1) andf (w) =

ji(−β/w; q)∞ Y 1 Y (sw; q)∞.
(−x1 w; q)∞ l=1 w − vl l=2 (−xl w; q)∞94(3.60)

Here C is a closed complex contour encircling 0, v1 , v2 , . . . and such that for all r ∈ d + iR,
C contains −q r+k β for all k ≥ 0, but leaves outside 1/(q r+k s) and 1/(q k xl ) for all k, l ≥ 1.
Proof. We use an analytic continuation argument staring from identity (3.54). Considering specializations sg(β) for v0 , q J0 and sg(x1 ), sqW(x1 ), sqW(x2 ), . . . , respectively for
u0 , q I0 , u1 , q I1 , u2 , q I2 , . . . , we can prove expression (3.58) for values β, vl , xl is a smallneighborhood of the origin. Once (3.58) is established for parameters in an open set,
we can perform an analytic continuation, always keeping them in a region where they
define a probability measure. This is possible since both sides of (3.58) can be written as
absolutely convergent series of holomorphic functions in β, vl , xl .
Using the integral expression for the q-moments (3.57) we can obtain an alternative
expression for the Fredholm determinant:
Theorem 3.5.7. Assume conditions (3.56). Let Ce be a closed positively oriented contoureencircling −1/x1 , −1/x2 , . . . and which does not contain any point of the interior of q C.
eThen the Fredholm determinantal formula (3.58) holds when replacing C with C.

Proof. The proof of this alternative representation can be given as in theorem 3.5.2. We
can start from step-Bernoulli boundary conditions and then obtain stationary boundary
conditions via fusion. For detail we refer to [BMP19]

Remark 3.5.8. Another Fredholm determinantal formula for the stochastic higher spin
six vertex model with two-sided stationary boundary conditions was obtained in [IMS20]
and we use this below in chapter 4. While the two formulas differ from each other, one
should in principle be able to transform one to the other. We do not focus on this in thepresent work.

3.5.3

4 φ3 stochastic vertex model observables.

By using the fact that the 4 φ3 vertex model is equivalent in distribution to a marginalof the sqW/sqW field we can obtain contour integral expressions for the q-moments of
the height function Hφul . Indeed, this is possible by employing the eigenoperator D1 .
However, only finitely many of the q-moments exist, and this also involves certain bounds
on the parameters. Consider the model with step-stationary boundary conditions (3.30).
Assume that x1 , x2 , . . . satisfy (3.56).
Proposition 3.5.9. If l is such that q l > max1≤t≤j {x1 yt }, we have
I
I


Y
zA − zBul (i,j)−l Hφstepl l(l−1)/2
Eq= (−1) q···Γ[x|1]
Γ[x|l] 1≤A<B≤l zA − qzB
(
)
j
l
i
Y
Y
1 + szk Y zk + s/q
dzk
×
,
1
−
x
z
z
−
y
/q
2πiz
(1
+
sz
)
α
k
k
α
k
k
α=1
k=1 α=1(3.61)

where Γ[x|j] is a positively oriented contour around 1/x1 , 1/x2 , . . . , qΓ[x|j+ 1], and noul (i,j)
−l Hφ
l
step
other pole of the integrand. In case q ≤ max1≤t≤y {x1 yt } we have E
q
= ∞.
95

Despite the fact that the distribution of Hφul is not characterized by its q-moments
since only finitely many of them exist, we can still write down Fredholm determinant
expressions for the q-Laplace transform of Hφul .
Theorem 3.5.10. Consider the 4 φ3 stochastic vertex model with two-sided stationary
boundary conditions with parameters (3.46). Let M ∼ q-Poi(x1 β) be independent of thevertex model. We have
E

1
(ζq

(3.62)(−ζ)r f(w)/f(q r w)
dr,
q r w − w0
d+iR sin(πr)(3.63)The kernel K is defined by
1
K(w, w ) =2i
0= det(Id + K)L2 (C) .

ul (i,j)−M
−Hφ

; q)∞

Z

where d ∈ (0, 1) andji(−β/w; q)∞ Y (−yl /w; q)∞ Y (sw; q)∞f(w) =(−x1 w; q)∞ l=1 (s/w; q)∞ l=2 (−xl w; q)∞(3.64)

Here C is a closed complex contour encircling 0, q k s for k ≥ 0 and such that, for any
r ∈ d + iR, C contains q r C and −q r+k yl , −q r+k β for all k, l ≥ 0, but leaves outside
1/(q r+k s) and −1/(q k xl ) for all k, l ≥ 0.
Proof. Expression (3.62) is derived from the general summation identity (3.54) in the same
way as Theorem 3.5.6. First we establish (3.62) for parameters β, s, yl , xl is a small neighborhood of the origin by considering specializations of u0 , q J0 , u1 , q J1 , . . . , v0 , q I0 , v1 , q I1 in(3.54). Subsequently we relax conditions on these parameters moving them away fromthe origin but keeping them in real intervals in such a way that they always define aprobability measure. This is possible due to the analyticity of both sides of (3.62) in theparameters.
e be a closed complex contour encircling −1/x1 , −1/x2 , . . .
Theorem 3.5.11. Assume (3.56) and let C
e Then expression (3.62) holds
and that does not contain any point of the interior of q C.
ewith contour C replaced by C.
Proof. This alternative determinantal expression for the q-Laplace transform follows from
Theorem 3.5.7 using the sqW specializations and subsequent analytic continuation.
Remark 3.5.12. Both Theorems 3.5.10 and 3.5.11 degenerate to Fredholm determinantal
formulas for the q-Hahn pushTASEP. In particular, expression given by Theorem 3.5.11
was conjectured in [CMP19] (Conjecture 3.11) for step initial conditions. Therefore, wehave established this conjecture. Moreover, by sending all parameters to 1, one canalso get the proof of [CMP19, Conjecture 4.6] on the Laplace transform of the one-point
observable in the beta polymer like model introduced in [CMP19].
96

Chapter 4
KPZ fluctuation of the Higher Spin
Vertex Model
This chapter presents results contained in sections 5 and 6 of
[IMS20] T. Imamura, M. Mucciconi, and T. Sasamoto. “Stationary Higher Spin Six Vertex
Model and q-Whittaker measure”. In: Probability Theory and Related Fields (Mar.
2020)

4.1

Fredholm determinants from elliptic determinants

In section 3.5 we derived a number of determinantal formulas for the higher spin vertex
model and its degenerations. There the scheme we employed was to use nice nested contours representations for q-moment formulas, that after a certain shift of contours give riseto terms of the series expansion of a Fredholm determinant. Nevertheless this is not the
only way such nice expressions can be derived. A possibly more straightforward procedure was discovered in [IS19] and makes use of Frobenius elliptic generalization of Cauchy
determinant (see [KN03]). In this section we will report an alternative representation for
the q-Laplace transform of the height function of the higher spin six vertex model foundin [IMS20] and we will omit the details of the derivation of our formulas, which can befound in [IS19; IMS20].

4.1.1

Double sided stationary higher spin vertex model

Techniques employing elliptic determinants allow to consider a more general vertex model
than the one discussed in chapter 3. In fact throughout the section we will allow spinparameter to be completely inhomogeneous in space. This requires a change of notation
the and stochastic weights we use are those reported in table 4.1. Moreover, since in thischapter we only consider the up-right higher spin vertex model and no other model wewill drop the use of superscripts ur and we will refer to the stochastic weight as Lur = L
ur
and to the height function HHS= H.
97

Lξx ut ,sx

g

g−1

g

g+1

g

g

g

g

1−q g ξx sx ut
1−sx ξx ut−sx ξx ut +q g ξx sx ut
1−sx ξx ut−sx ξx ut +s2x q g
1−sx ξx ut

1−s2x q g
1−sx ξx ut

Table 4.1: Fully inhomogeneous stochastic weights
Definition 4.1.1 ([BP18a]). Specialize the up-right stochastic vertex model of Definition 3.3.1 by taking Lur(x,t) = Lξx ut ,sx , where the latter are given in table 4.1. We refer tothis model as the fully inhomogeneous stochastic higher spin six vertex model. To ensure
well posedness of the model we assume that parameters are
0 ≤ q < 1,

0 ≤ sx < 1,

ξx > 0,ut < 0,for all x, t.

(4.1)

We consider the two sided-stationary boundary conditions:bhx ∼ qNB(s2x , v/(ξx sx )),bht ∼ Ber(d ut /(d ut − 1)),

(4.2)

where Ber(·) are independent Bernoulli random variables and qNB(·) are independent
q-negative binomial random variables (A.18). Parameters v, d play the role of densities
and we assume
q sup{ξi si } < d < inf {ξi si } ≤ sup{ξi si } < inf {ξi /si }(4.3)

0 ≤ v < inf {ξx sx }.

(4.4)

i

i

i

i

and
x

It was proven in [IMS20] that even for a fully inhomogeneous model such as that of definition 4.1.1 a notion of translation invariance and hence stationarity can be defined. Forh= (bhi+1 , bhi+2 , . . . ),this consider random boundary conditions B h , B v and their shifts B+iv
B+j= (bvj+1 , bvj+2 , . . . ). We say that a measure P(B h , B v ) is translation invariant if therestriction of the ensemble to the quadrant Z≥i × Z≥j originating from a generic pointhv(i, j) is equivalent in distribution to P(B+i, B+j).
We have the following
Theorem 4.1.2 ([IMS20]). Consider the higher spin six vertex model of definition 4.1.1
with double sided-stationary boundary conditions of densities v, d . Then, if v = d themodel is transfation invariant and we call it stationary higher spin vertex model withdensity d .

4.1.2

Fredholm determinants for two sided stationary boundary
conditions: elliptic determinants form

In this section we give a Fredholm determinant expression for the q-Laplace transform of
the probability mass function of the shifted height function H(x, t)−M. The proof of theorem 4.1.3 is based on calculations involving an elliptic version of the Cauchy determinantthat were developed in [IS19] and it is therefore omitted.
98

Theorem 4.1.3. Consider the higher spin vertex model of definition 4.1.1 and let v < d .
Also set M to be an independent q-Poisson random variable of parameter v/d (see (A.19)
for the definition). Then we have1
= det(1 − f K)l2 (Z) .
(4.5)
EHS(v,d )⊗M(ζq H(x,t)−M ; q)∞
where
f (n) =
K(n, m) =

x−1
X
l=1

φl (n) = τ (n)Z

1,
1 − q n /ζ(4.6)

φl (m)ψl (n) + (d − v)Φx (m)Ψx (n),lY11
(qv/w; q)∞dw,
x+n−l+1
(w
−
ξ
F
(w)
k+1 sk+1 )D 2πi wk=1

Zl
ξl+1 sl+1
dz n+x−l−1 Y
F (z)ψl (n) =,z(z − ξk sk )
τ (n) C 2πi(qv/z;
q)
∞
k=2
Z
x
dw 1
1 Y
1
(qv/w; q)∞Φx (n) = τ (n),n+1
w − d k=2 w − ξk sk F (w)D 2πi wZx
1
dz z n−1 Y
Ψx (n) =(z − ξk sk )F (z).
τ (n) C 2πi (v/z; q)∞ k=2

(4.7)

(4.8)

(4.9)

(4.10)(4.11)

The contour D encircles {d , ξ2 s2 , . . . , ξx sx } and no other singularity, whereas C contains
0 and vq k , for any k in Z≥0 . Finally, τ (n) is taken to be(bn
if n ≥ 0,
τ (n) =
(4.12)cn
if n < 0,withv < b < d ≤ inf {ξi si } ≤ sup{ξi si } < c < inf {ξi /si },
i≥2

and

i≥2

i≥2

t
x
Y
Y
(qz/(ξk sk ); q)∞
F (z) = (qz/d ; q)∞ (1 − zuj ).
(zsk /ξk ; q)∞j=1
k=2(4.13)The next proposition ensures that the Fredholm determinant expression (4.5) makes
sense.
Proposition 4.1.4. The kernel f K defined by eqs. (4.6) to (4.11) is trace class.

99

Proof. From f K being a finite sum of products of operators of rank one, it is enoughto show that each one of these operators is of Hilbert-Schmidt class. This is essentiallyproven in Appendix C.1. In fact the generic terms f (n)φl (n)ψl (m) and f (n)Φx (n)Ψx (m)are bounded in absolute value by quantities exponentially small in |n| + |m|, thanks to
Proposition C.1.1, and therefore the double summationX|f (n)K(n, m)|2n,m∈Z

is indeed convergent.
An alternative expression for the kernel K is given defining an auxiliary kernel A as
A(n, m) =x−1Xφl (n)ψl (m)(4.14)

l=1

of which we report the explicit form.
Proposition 4.1.5 (Double integral kernel). The discrete kernel A admits the following
expressiont 
z m Y 1 − uj z
dw
dz n+1w
1 − uj w
D
C
j=1

x 
Y
(z/(ξk sk ), wsk /ξk ; q)∞ (qv/w, qz/d ; q)∞ 1
.
×
(w/(ξ
(qv/z,
qw/d;
q)k sk ), zsk /ξk ; q)∞
∞z −w
k=2

τ (n) 1A(n, m) =τ (m) (2πi)2ZZ(4.15)

Proof. All it takes to show (4.15) is to perform the summationx−1X

φl (n)ψl (m),l=1using the rather tricky identity
"
# x−1
x
l
X al+1 wl−1 Y1wx−1 Y z − akz − ak
−1 =
,
x−1
l
z−w z
w
−
a
w
−
a
w
−
a
k
l+1 z
k
k=2
l=1
k=2
which can be proven by induction. We see that the addend (z − w)−1 in the left handside doesn’t give any contribution to the integral as integrating over the variable z it only
leaves an integral in w over a path containing no singularities.
We like now to state some regularity properties of the integral kernel f A defined by(4.6),(4.14) that hold for parameters v, d such thatqv < d < v/q.
100

(4.16)

Proposition 4.1.6. Let ζ < 0 and take f, A as in (4.6), (4.14). Then 1 − f A is aninvertible operator. Moreover both f A and (1 − f A)−1 are well defined, bounded operatorsin the region (4.16).
In the proof of Proposition 4.1.6 we use the following biorthogonality property.
Lemma 4.1.7. Let v, d be such that v < d or (4.16) holds. Then we haveX
φl (n)ψm (n) = δl,m .

(4.17)

n∈Z

Proof. This is a simple consequence of the contour integral expressions (4.8), (4.9). The
generic term in the summation in the left hand side of (4.17) isQ
Z Z
dwdz  z n wl−1 m
k=2 (z − ξk sk ) G(w),φl (n)ψm (n) = ξm+1 sm+1
Ql+1
2
m+1
w z
G(z)
D C (2πi)k=2 (w − ξk sk )where, in the function G, we gathered together the factors independent on n or l, m as
G(u) =(qv/u; q)∞.
ux F (u)We see that in order to take the summation over all integers inside the integrals we need|z/w| to be suitably defined depending on the positivity of n itself.
e being a circle of radius r such that maxi |ξi si | < r < mini |ξi /si |.
Consider the contour C
We can write"
X Z Z dwdz  z n wl−1 Qm (z − ξk sk ) G(w)X
k=2
φl (n)ψm (n) = ξm+1 sm+1
2
m+1 Ql+1
(2πi)
w
z
G(z)
D
C
k=2 (w − ξk sk )
n≥0
n∈Z
#
X Z Z dwdz  z n wl−1 Qm (z − ξk sk ) G(w)k=2+2m+1 Ql+1
(2πi)
w
z
G(z)
e
k=2 (w − ξk sk )n<0 D C
"Z Z
#
Qm
Z 
(z
−
ξ
s
)
G(w)
dwdz 1wl
k k
k=2= ξm+1 sm+1−
2 z − w z m+1 Ql+1
(2πi)
G(z)
e
D
C
C
k=2 (w − ξk sk )
#
"Z Z
Qm
wl
dwdz 1k=2 (z − ξk sk ) G(w),= ξm+1 sm+1
Ql+1
2
m+1
G(z)
e (2πi) z − w z
(w
−
ξ
s
)
D D
k
k
k=2
e into an union or the two contours C and D,ewhere, for the last equality, we deformed Cwith the latter being a curve encircling D and no other singularity for the z variable.
Performing the z integral we get
!
Z
l+1
m
Y
Y
1
dw l−m−1
1m=l+1 + 1m≤l
ξm+1 sm+1w+ 1m≥l+2(w − ξk sk ) .
w
−
ξ
k sk
D 2πi
k=m+1
k=l+2

Naturally, when m > l there is no pole inside D and the integral vanishes. On the other
hand, if m ≤ l we can evaluate the residue at infinity and obtain the result.
101

Proof of Proposition 4.1.6. We want to show that kf Akl2 (Z) < 1, so to define (1 − f A)−1
through the geometric seriesX(f A)k .
k≥0

A first observation is that kAk = 1 and this follows from the biorthogonality relation(4.17). For this set V = span{ψi | i = 1, . . . , x − 1}P
and notice that, by biorthogonality,
we also have V = span{φi | i = 1, . . . , x − 1} (if ve =ci φi such that ve ⊥ V , then ve = 0).
2
For any h ∈ ` (Z) write the orthogonal decomposition h = hV + hV ⊥ where hV ∈ V andhV ⊥ ∈ V ⊥ . Then we have Ah = AhV = hV for all h since Aφi = φi for all i = 1, . . . , x − 1
and we conclude thatkAhV k= 1.
kAk = suphV ∈V khV k
To show that the operator norm of f A is strictly smaller than 1, consider the following
bound for the function f , defined in (4.6),
f (n) ≤ (1 − ε)f (n + n0 ) + εPn1 (n),with Pn1 (n) = 1n≥n1 and ε, n0 , n1 , suitably chosen. In particular, taking n0 large enoughand ε small we can let n1 be an arbitrary big number. Moreover, the fact that f is adiagonal operator implies the simple estimate
kf Ak ≤ (1 − ε)kf kkAk + εkPn1 Ak ≤ 1 − ε + εkPn1 Ak.
Let’s consider now an element η in the unitary sphere of l2 (Z). By simply using thedefinition of the kernel A and the Schwartz inequality we have


!1/2
2 1/2
x−1
x−1
X
X
X
X
X

kPn1 Aηk ≤1n≥n1 φl (n)ψl (m)η(m)  ≤kψl k|φl (n)|2
,
l=1

n∈Z

m∈Z

l=1

n≥n1

which, thanks to the bound (C.1), can be shown to be geometrically small in n1 . Therefore
kf Ak < 1.
Finally, we remark that for sequences φl or ψl the bounds (C.1) hold true also in theregion (4.16), so A is analytic in this domain, so are its powers and so is (1 − f A)−1 asone can show the geometrical decay of derivatives of (f A)N as well.

4.1.3

Decoupling and stationary limit

In this Section we give a proof of Theorem 4.1.8, that characterizes the probability distribution of the height function H in the higher spin vertex model with double sided
stationary boundary conditions. Our starting point is the Fredholm determinant formula
stated in Theorem 4.1.3. Removing the effect of the independent q-Poisson random variable M from expression (4.5) we find that determinantal expressions we obtain are welldefined in the region (4.16). In Corollary 4.1.10 we specialize the result of Theorem 4.1.8
to the relevant case of the stationary higher spin vertex model with density d .
102

Theorem 4.1.8. Consider the higher spin vertex model of definition 4.1.1 and assumev, d satisfy (4.16). Then we haveE1
(ζq H(x,t) ; q)∞X (−1)k q (k2)  v k1=
Vx;v,d (ζq −k ),(qv/d ; q)∞ k≥0 (q; q)kd(4.18)

where the function Vx;v,d is defined as
Vx;v,d (ζ) =1det(1 − f K)l2 (Z) .
1 − v/d

The following lemma offers a tool to decouple a generic process from the contributionof an independent q-Poisson random variable.
Lemma 4.1.9. Let M ∼ qPoi(p). Then, for any bounded function B, we havek
1 X (−1)k q (2) k
B(z) =
p EM (B(z − M − k)).
(p; q)∞ k≥0 (q; q)k(4.19)

Proof. To verify identity (4.19) we simply open up the average in the right hand side withrespect to M, as
EM (B(z − M − k)) =

X
l≥0

pl

(p; q)∞
B(z − l − k).
(q; q)l

We can now rearrange the double summation in the indices k, l, naming L = l + k, asLXk(−1)k q (2)rhs of (4.19) =B(z − L),(q; q)k (q; q)L−k
L≥0
k=0
X

which completes the proof, after recognizing, in the right hand side, the q-Pochhammer
expansion (A.3) (with z = 1) that is one for L = 0 and zero otherwise.
In the remaining part of the paper we will use the following decomposition of terms
Φx , Ψx :(2)
Φx (n) = Φ(1)x (n) + Φx (n),(4.20)(2)
Ψx (n) = Ψ(1)x (n) + Ψx (n),(4.21)

obtained separating from the integration (4.10) (resp. (4.11)) the contribution of polew = d (resp. z = v) from that of other poles. The exact expressions are
τ (n)
Φ(1)
x (n) =
n+1
d

x
Y

1
(qv/d ; q)∞,d − ξk sk F (d )k=2103

(4.22)

Φ(2)x (n) = τ (n)

Z

x
dw 1
1 Y
1
(qv/w; q)∞,n+1
w − d k=2 w − ξk sk F (w)D1 2πi w(4.23)xΨ(1)x (n) =
Ψ(2)x (n) =

1
τ (n)

Z

vn Y
F (v)
,
(v − ξk sk )
τ (n) k=2(q; q)∞(4.24)x
dz z n−1 Y(z − ξk sk )F (z),C1 2πi (v/z; q)∞ k=2(4.25)

where F was given in (4.13), contour D1 contains {ξi si }i≥2 and no other singularity andC1 contains {q k v}k≥1 and no other singularity.
Proof of Theorem 4.1.8. Using Lemma 4.1.9 setting B : z → E(1/(ζq H+z ; q)∞ ) and expressing the q-Laplace transform EHS(v,d ) (1/(ζq H−M ; q)∞ ) as in (4.5) we obtain formula(4.18). Therefore we only need to show that expression (4.18) is well posed in the region(4.16).
Using basic properties of Fredholm determinants, along with the regularity of thekernel f A proved in Proposition 4.1.6, we can writedet(1 − f K) = det (1 − f A − (d − v)f Φx Ψx ) = det(1 − f A) det (1 − (d − v)(%f Φx )Ψx ) ,where we called % = (1 − f A)−1 . This allows us to express Vx;v,d (ζ) as!

(%f Φx )(n)Ψx (n) ,(4.26)

We turn our attention to the term
X
XX(%f Φx )(n)Ψx (n) =f (n)Φx (n)Ψx (n) +(f A%Φx )(n)Ψx (n),(4.27)

1
Vx;v,d (ζ) = det(1 − f A)
1 − v/dn∈Z

1 − (d − v)

n∈Z

X
n∈Z

n∈Z

that we rewrite, using eqs. (4.22) to (4.25) as
X
X X
X
(1)
(j)
f (n)Φ(1)f (n)Φ(i)(f A%f Φx )(n)Ψx (n).
(4.27) =x (n)Ψx (n)+x (n)Ψx (n)+n∈Zi,j=1,2 n∈Z(i,j)6=(1,1)

n∈Z(4.28)
We easily see that, thanks to bounds stated in Appendix C.1, the second and the thirdterms in the right hand side of (4.28) are geometrically convergent summations in theregion (4.16). On the other hand, the generic term of the summation in the first addendof the right hand side of (4.28) takes the form
1
1  v n Y v − ξk sk (qv/d ; q)∞ F (v).
1 − q n /ζ d dd − ξk sk (q; q)∞ F (d )
k=2
x

104

We can perform the summation over n through the Ramanujan 1 ψ1 formula ([Ism05],
Theorem 12.3.1) as v n (v/(d ζ), qζd /v, q, q; q)
1
∞
.
=
n /ζ
1
−
q
d
(v/d
,
qd
/v,
1/ζ,
qζ;
q)
∞
n∈Z

X

which itself leads to the expressionXn∈Z

1 (v/(d ζ), qζd /v, q; q)∞ F (v)(1)
f (n)Φ(1)x (n)Ψx (n) =d (v/d , 1/ζ, qζ; q)∞ F (d )xYv − ξk sk

d − ξk sk
k=2.

(4.29)

Combining the explicit formula (4.29) with (4.28) and (4.26) we see that Vx;v,d does notin fact present any singularity in v = d by virtue of the Taylor expansion
!
t
X
1
1
+
rhs of (4.29) =υ0 (1/ζ) − υ0 (qζ) + 2υ0 (q) + xh0 (d ) −a0 (d ; j) +O (d − v) ,d −v dj=1(4.30)
where we used the q-polygamma type functions υk defined in Appendix A, their combinations a0 , h0 presented in (4.34) and the notation a0 (d ; j) stresses the dependence on the
spectral parameters uj asak (d ; j) = υk (quj d ) − υk (uj d ).
Therefore Vx;v,d is an analytic function of both parameters v, d in the region (4.16) andthis concludes the proof.
Calculations performed during the proof of Theorem 4.1.8 can be exploited to obtain
determinantal formulas for for the stationary higher spin vertex model.
Corollary 4.1.10. Consider stationary higher spin vertex model as in definition 4.1.1.
Then we havek
1 X (−1)k q (2) k
1=
EHS(d ,d )q Vx (ζq −k ) − Vx (ζq −k−1 ) , (4.31)
H(x,t)(ζq; q)∞(q; q)∞ k≥0 (q; q)kwith the function Vx = Vx;d ,d being
Vx (ζ) = det (1 − f A)

− υ0 (1/ζ) + υ0 (qζ) − 2υ0 (q) − xh0 (d ) +
−d

X

X

i,j=1,2 n∈Z(i,j)6=(1,1)

f (n)Φx(i) (n)Ψ(j)x (n) − d

t
X
j=1

X
n∈Z

a0 (d ; j)!

(f A%f Φx )(n)Ψx (n) .
(4.32)

105

Proof. All it takes to show (4.31) is to take the limit v → d of both sides of equality(4.5). We exchange the limit sign in v → d and the summation sign in the right hand sideof (4.5) and this can be done as, in the proof of Theorem 4.1.8, the function Vx;v,d wasshown to be uniformly bounded in a neighborhood of v = d and its limit Vx;v,d → Vx;d ,dis readily computed using (4.30). This is enough to prove thatk
1 X (−1)k q (2)1=
Vx (ζq −k ).
(4.33)
lim EHS(v,d )v→d(ζq H(x,t) ; q)∞(q; q)∞ k≥0 (q; q)kFinally, substituting Vx (ζq −k ) with q k Vx (ζq −k ) + (1 − q k )Vx (ζq −k ) and rearranging thesummation in the right hand side of (4.33) we obtain (4.31).

4.2

Asymptotic analysis in the stationary regime

In this section we prove that the asymptotic fluctuations of the stationary higher spinvertex model obey the Baik-Rains distribution, when taken along a special direction. For
simplicity we will suppress time inhomogeneities and we will take the model in definition 4.1.1 with ut = u and v = d .
We first fix a number of scaling parameters and for this make use of q-polygamma typefunctions υk defined in Appendix A. For non-negative integers k consider the functionsx

ak (d ) = υk (qud ) − υk (ud ),1X(υk (d /(ξy sy )) − υk (d sy /ξy ))hk (d ) =x y=2

and, depending on the parameter d , define the quantities1/3
1h1 (d )
κ0 =,
η0 = κ0 a0 (d ) − h0 (d ),γ=−(κ0 a2 (d ) − h2 (d )).
a1 (d )
2

(4.34)(4.35)

We assume that the functions hk always converge in the large x limit and we refer tothe curve (x, κ0 x) as the characteristic line of the stationary higher spin vertex model.
For random growth models usually the characteristic line is expressed as a function of thetime t, rather than of the coordinate x, but in our case, since the system exhibits spatial
inhomogeneities we find more natural to adopt the notation (x, κ0 x). The parameter η0multiplied by x is readily understood as the expectation E(H(x, κ0 x)), whereas γ will be
used to describe the size of the characteristic fluctuations of H around η0 . By slightly
perturbing quantities κ0 , η0 we can analyze the asymptotic behavior of H in a region ofsize x2/3 around the characteristic line. For this we extend the definitions given in (4.35)
settingh2 a1 − h1 a2 $(4.36)
κ$ = κ0 +a21
γx1/3
a0 (h2 a1 − h1 a2 ) $h2 a1 − h1 a2 $2
η$ = η0 ++,(4.37)
a21
γx1/3
a1
γ 2 x2/3
where $ is a real number parameterizing the displacement from the characteristic line
and functions ak , hk are evaluated at d . We come now to state our main result.
106

Theorem 4.2.1. Consider the stationary higher spin vertex model with density d asin theorem 4.1.2. Assume that parameter satisfy additional conditions given below in
definition 4.2.5. Then, for any real numbers $, r we haveH(x, κ$ x) − η$ x
lim PHS(d ,d )> −r = F$ (r),(4.38)x→∞γx1/3
where F$ (r) is the Baik-Rains distribution presented below in Definition 4.2.3.

4.2.1

The KPZ scaling for the Higher Spin Six Vertex Model

Before we enter the discussion it is appropriate to recall the definition of the Baik-Rains
distribution [BR00], which will ultimately describe long time fluctuations of the stationary
height function H under a suitable scaling. Rather than showing the original definition
given by authors in [BR00], formula (2.16), we present an equivalent expression first foundin [IS13] (see also [FS06]).
The building blocks for the construction of the Baik-Rains distributions are given in
the following
Definition 4.2.2 (Airy function and Airy kernel). The Airy function Ai is given by 3

Z e 3i π ∞z
1exp− zν dz,
Ai(ν) =
2πi e− 3i π ∞3iwhere the integration contour is any open complex curve having the half lines {Re 3 π |R ≥i0} and {Re− 3 π |R ≥ 0} as asymptotes.
The Airy kernel1 KAiry is defined as
 3

Z −πi
Z e 32 πi ∞dz e 3 ∞ dw 1
w
z3
exp
−
− wν + zθ ,(4.39)
KAiry (ν, θ) =2π2πi z − w33
e− 3 πi ∞ 2πi e 3 i ∞where again integration contours are non intersecting complex curves whose asymptotes
2iare half lines {Re± 3 π |R ≥ 0} for w and {Re∓i 3 π |R ≥ 0} for z.
We come to the next

Definition 4.2.3 (Baik-Rains distribution). Let $ ∈ R and define the one parameterfamily of functions
X Z ∞ (i)2
χ$ (r) = F2 (r) r − $ −Υ−$ (ν)Υ(j)$ (ν)dνi,j=1,2(i,j)6=(1,1)

−

Z ∞
r

r

dνΥ$ (ν)

Z ∞

dλ1

r

Z ∞
r

dλ2 %Airy;r (ν, λ1 )KAiry (λ1 , λ2 )Υ−$ (λ2 )!

(4.40)(i)where terms F2 , %Airy;r , Υ$ , Υ$ are given by:
1

Sometimes the equivalent expression KAiry (ν, θ) =

107

R∞
0

Ai(λ + ν)Ai(λ + θ)dλ is found in literature.

• F2 (r) is the GUE Tracy-Widom distribution ([TW94])

F2 (r) = det 1 − 1[r,∞) KAiry L2 (R) ;• %Airy;r (ν, λ) is the kernel%Airy;r (ν, λ) = 1 − 1[r,∞) KAiry
(1)

(2)

• auxiliary functions Υ$ , Υ$ are
Υ(1)$ (ν) = e

$3
−ν$
3

Υ(2)
$ (ν) =,

−1

Z e 23 iπ ∞(4.41)(ν, λ);(4.42)

3dω exp{− ω3 + ων},2
ω+$e− 3 iπ ∞ 2πi(4.43)

where the integration contour passes to the left of −$;• lastly(2)
Υ$ (ν) = Υ(1)$ (ν) + Υ$ (ν).

(4.44)The Baik-Rains distribution F$ is
F$ (r) =∂χ$ (r).
∂r(4.45)

Remark 4.2.4. The equivalence between F$ of (4.45) and an analogous expression foundin [FS06] is discussed in the Remark at the end of Section 5 of [IS19]. Further, in Appendix A of [FS06] the authors prove that their definition of the Baik-Rains distribution
is equivalent to the original one introduced in [BR00].
We now possess all the ingredients to give a brief explanation of the KPZ scaling
theory, which gives a precise conjecture to describe stationary (asymptotic) fluctuationsof the height function of models in the KPZ universality class ([PS00]).
We start considering a properly rescaled version of our model, where, for convenience
we interpret the vertical spatial coordinate as a time direction and where we regard
space and time as continuous parameters. In this case the height function H, defined forthe higher spin vertex model in (3.21), still contains every information on the random
dynamics thanks to relations
H(x, t) − H(x + dx, t) = # of paths in [x, x + dx] at time t,(4.46)
H(x, t + dt) − H(x, t) = # of paths crossing x during the time interval [t, t + dt].
(4.47)For the sake of argument assume that the average of space and time infinitesimal increments of H are regular enough to define the deterministic density ρ and current jas
E (H(x + dx, t) − H(x, t)) ≈ −ρ(x, t)dx,
108

E (H(x, t + dt) − H(x, t)) ≈ j(x, t)dt.

The system is autonomous, or, in other words, its evolution depends on space and time
only implicitly, therefore the current j must only be a function of ρ and the continuity
equation linking these quantities reads∂t ρ(x, t) + ∂x j(ρ(x, t)) = 0.

(4.48)At this stage the height H remains defined, through (4.46), (4.47), only up to a global
constant and to remove this ambiguity we fix its value at the reference point x = 0, t = 0
to be H(0, 0) = 0. With this choice the average profile of H at the generic space time
point (x, t) can be expressed as
Z t
Z xρ(y, 0)dy +j(ρ(x, s))ds(4.49)
η(x, t) = −00

and the study of fluctuations of the height is, by definition, the study of the random
quantity
H(x, t) − η(x, t).
(4.50)
Assume now that the system has reached its steady state, or equivalently assume thatat time zero the measure is stationary. Qualitatively, the randomness of (4.50) is affectedby two different contributions. One is coming from the stochastic evolution of the systemand the other is given by initial conditions. For growth processes in the KPZ universality
class, when initial conditions are deterministic and sufficiently regular, fluctuations in thelong time scale are expected to present with size of order t1/3 . This conjecture goes backto the seminal paper [KPZ86], where authors argued such property to hold for the solutionof the one dimensional KPZ equation itself. On the other hand, from our knowledge ofthe stationary measure of the higher spin vertex model displayed in theorem 4.1.2, we
certainly expect fluctuations in the space direction to have size of order x1/2 , as a resultof the Central Limit Theorem applied to independent occupation numbers at each site.
This means that the information we have about H, which is the choice H(0, 0) = 0,
will be transported by the random dynamics along the direction of growth of the surface
and along this line we can observe the emergence of the 1/3 exponent. Along all otherlines, the distribution of (4.50) will be affected very little by the process, and asymptotic
fluctuations remain of gaussian nature. An earlier evidence of this last fact was found in[FF94] (see also [BFP10], Appendix D).
The direction along which nontrivial fluctuations are observed is given by the characteristic line of partial differential equation (4.48). This is the curve (xt , t), where xt is set
to be the solution of the differential equation(ẋt = j 0 (ρ(xt , t)),(4.51)x0 = 0.
When the system is in its stationary state, equation (4.51) loses its dependence on time
and the ẋt is only function of the stationary density profile ρst . In case the model does
not present space inhomogeneities the characteristic curve is simply the line (j 0 (ρst )t, t),109

but when the stationary density is not constant, this is no more true. We use the explicit
parametrization (x, tx ), rather than (xt , t) and by integration of (4.51), we obtain
Z xdy.
(4.52)tx =0
0 j (ρst (y))

To reiterate what we just explained, consider diverging x and t. Assume first that|t − tx | = O(x2/3+δ ),for some δ > 0. Then, asymptotically (4.50) obeys the gaussian distributions and its
size becomes of order x1/2 . When on the other hand, (x, t) is taken in the vicinity of thecharacteristic curve, say|t − tx | = O(x2/3 ),then fluctuations become of size x1/3 and their law is described by the Baik-Rains distribution.
We can be more precise. Take at first t = tx . The convergence result, in this case, isH(x, tx ) − η(x, tx ) D−−−→ F0 ,
x→∞
γx1/3

where, calling σy2 dy the variance of the number of paths lying at time 0 in the infinitesimal
segment [y, y + dy] and its mean
Z
1 x 2
2
σ = limσy dy,
x→∞ x 0then the constant γ is given by

1 00
tx
j (ρst )(σ 2 )2 .
x→∞ 2x

γ 3 = − lim(4.53)The explicit parametrization of a fan of size x2/3 around the characteristic line can be still
expressed in terms of macroscopic quantities. Consider a perturbation of tx of the form
tx,$ = tx − $σ 2 j 00 (ρst ) 2/3
x .
γj 0 (ρst )2(4.54)

with $ being a real number. The resulting effect on the expression of η reads, up to order
x1/3 , as
σ 2 j 00 (ρst )j(ρst ) 2/3 1 2 (σ 2 )2 j 00 (ρst ) 1/3
x − $
x .
(4.55)
ηx,$ = η(x, tx ) − $γj 0 (ρst )22
γ 2 j 0 (ρst )In this case, the convergence result for fluctuations along the line (x, tx,$ ) becomesH(x, tx,$ ) − ηx,$ D−−−→ F$ .
x→∞
γx1/3

(4.56)The same kind of results are conjectured for discrete time systems, where the characteristic curves can be again explicitly expressed through relation (4.52). In the next
section we will establish result (4.56) for the stationary higher spin vertex model. For thismodel, the scaling parameters κ$ , η$ , γ were defined in eqs. (4.35) to (4.37) and it is a
simple exercise to verify that they match with expressions given in eqs. (4.53) to (4.55).
110

4.2.2The Baik-Rains limit

This subsection is devoted to the proof of Theorem 4.2.1, that characterizes the asymptotic
fluctuations of the height function in the stationary higher spin vertex model. Throughout
the proof we will assume that the model presents only spatial inhomogeneities and hence
the spectral parameters U are taken as
U = (u, u, u, . . . ).

(4.57)

Our strategy relies on taking the large x limit of the q-Laplace transform of the heightfunctions H given in (4.31). The computation of such a limit for the right hand sideof (4.31) might look complicated, but the expression will simplify after the right changeof variables (discussed in detail below in Section 4.2.3). Motivated by the KPZ scaling
theory discussed in Section 4.2.1, we fix parameters t and ζ ast = κxand

1/3 r

ζ = −q −ηx+γx.

(4.58)

Here and in the rest of the Section, for the sake of a cleaner notation, we setη = η$ ,andκ = κ$ ,dropping the explicit dependence on the real number $ from η$ , κ$ introduced in eqs. (4.36)
and (4.37). The first choice in (4.58) means that we are considering the behaviour of theheight along the critical line, while the choice for ζ reflects the fact that we study fluctuations of size γx1/3 around the expected value η of the height function. We assume
the parameter r of (4.58) to be fixed throughout the entire section and it will ultimately
represent the argument of the Baik-Rains distribution, as in (4.38).
The asymptotic analysis of expressions given in Corollary 4.1.10 will be performed via
a rigorous steep descent method. Although we postpone the details of the analysis to
Subsections 4.2.4 and 4.2.5, we now fix hypothesis on parameters q, Ξ, S, that will holdtrue throughout the rest of the Section.
Definition 4.2.5 (Conditions on parameters). Take a, σ such that a > d and σ ∈ [0, 1).
Parameters q, d , Ξ, S are assumed to satisfy (4.1),(4.3) and they are spaced so that there
exist Ra , Rσ , Rq , with the properties thata ≤ ξk sk ≤ a + Ra ,σ ≤ s2k ≤ σ + Rσ ,for all k,

0 ≤ q ≤ Rq

(4.59)

and2a< d /q.
(4.60)
1+σ
Numbers Ra , Rσ , Rq are strictly positive, yet small in the sense given by Proposition C.2.3.
a + Ra <The first and the second conditions in (4.59) are not too much prohibitive and theyessentially say that we can consider perturbations of a general homogeneous model, sinceparameters a and σ can be chosen freely. The strongest assumption in Definition 4.2.5
111

is indeed the third one in (4.59) and it says that the parameter q has to remain reasonably close to 0. The reason for such restrictions in the choice of parameters lies in theperturbative approach we used to prove Proposition C.2.3. There, we showed the steep
ascent property (for the function g defined below in (4.80)) of integration contour D, inthe case where q = 0, ξk sk = a, s2k = σ for each k ≥ 2. Subsequently, through a continuity
argument we concluded that the same property must hold also when parameters are taken
in suitably small neighborhoods of our original choices, hence (4.59).
Although numerical checks show that a steep ascent contour D indeed exists also forq reasonably greater than zero (yet not too much close to 1), obtaining sharp bounds
for parameters becomes difficult due to the complicated expressions we encounter setting
q > 0. More precisely, to prove Theorem 4.2.1 when q is taken far from 0 would mean
constructing an explicit closed contour D on which one would be able to show that thefunction g assumes a global minimum in a neighborhood of d . This is indeed possible inprinciple, but obtaining explicit bounds for parameters q, ξk , sk becomes prohibitive.
The first inequality in (4.60) is also technical and not very restrictive. It is used in
the construction of the explicit steep ascent contour D in Proposition C.2.3. On theother hand the assumption 2a/(1 + σ) < d /q, reported in (4.60), is used to ensure theexponential decay of rear tails of f Φx Ψx in Propositions 4.2.9, 4.2.10.
Remark 4.2.6. Conditions stated in Definition 4.2.5 are far from being optimal andthey are essentially consequences of our choice for the representation of the integral kernels
K, Φx , Ψx in (4.7),(4.10),(4.11). In particular these technical assumptions are consequenceof the fact that D is a closed contour.
We will now present limiting expressions of terms entering the definition (4.32) offunction Vx (ζ), for x → ∞. The proof of the next Proposition is reported in Section 4.2.4.
Proposition 4.2.7. We havedet (1 − f A)l2 (Z) = F2 (r) +1
Rx(1) (r),1/3
γx(4.61)(1)

where F2 is defined in (4.41) and the error term Rx satisfies the following properties
1. For each r∗ ∈ R, there exists Mr∗ > 0 such that, for all x,
Rx(1) (r∗ ) < Mr∗ ;2. There exist  > 0, such that, for all r∗ ∈ [r − , r], we have


1
(1) ∗
(1) ∗
lim Rx (r ) − Rx (r − 1/3 ) = 0,x→∞γx(4.62)(4.63)

uniformly.
Let’s now see what is the asymptotic behavior of remaining terms of (4.32). The
proofs of the following three Propositions are given in Section 4.2.5 below.
112Proposition 4.2.8. Recall choice (4.58). Then we have1
1
(ta0 (d ) − υ0 (1/ζ) − 2υ0 (q) + υ0 (qζ) − xh0 (d )) = r − $2 + 1/3 Rx(2) (r). (4.64)
1/3
γx
γx
(2)

The error term Rx satisfies the following properties
1. for each r∗ ∈ R, there exists Mr∗ > 0 such that, for all x,
Rx(2) (r∗ ) < Mr∗ ;(4.65)

2. there exist  > 0, such that, foe all r∗ ∈ [r − , r] we have


1
(2) ∗
(2) ∗
lim Rx (r ) − Rx (r − 1/3 ) = 0.
x→∞
γx(4.66)

uniformly.
(i)

(j)

Lastly we state the convergence result for terms Φx , Φx , Ψx , Ψx .
Proposition 4.2.9. We have
d X X
(j)
f (n)Φ(i)x (n)Ψx (n) =γx1/3 n∈Z i,j=1,2(i,j)6=(1,1)

X

i,j=1,2(i,j)6=(1,1)

Z ∞
r

(i)

Υ−$ (ν)Υ(j)$ (ν)dν +1
R(3) (r),γx1/3 x(4.67)(1)(2)

where functions Υ , Υlowing properties(3)
are defined in (4.43) and the error term Rx satisfies the fol-

1. for each r∗ ∈ R there exists Mr∗ > 0 such that, for all x,
Rx(3) (r∗ ) < Mr∗ ;2. there exists  > 0, such that, for all r∗ ∈ [r − , r] we have


1
(3) ∗
(3) ∗
lim Rx (r + 1/3 ) − Rx (r ) = 0.
x→∞γx(4.68)(4.69)

uniformly.
Proposition 4.2.10. We haved X
(f A%f Φx )(n)Ψx (n) =γx1/3 n∈Z
Z ∞
Z ∞
Z ∞
dνΥ$ (ν)
dλ1
dλ2 %Airy;r (ν, λ1 )KAiry (λ1 , λ2 )Υ−$ (λ2 ) +
r

r

r

1
R(4) (r),γx1/3 x(4.70)

where the integral kernel %Airy;r and the function Υ$ were defined in (4.42), (4.44) and(4)
the error term Rx satisfies the following properties113

1. for each r∗ ∈ R there exists Mr∗ > 0 such that, for all x
Rx(4) (r∗ ) < Mr∗ ;(4.71)

2. there exists  > 0 such that, for all r∗ ∈ [r − , r], we have


1
(4) ∗
(4) ∗
lim Rx (r ) − Rx (r − 1/3 ) = 0.
x→∞γx(4.72)

uniformly.
Using convergence results reported in the Propositions 4.2.7, 4.2.8, 4.2.9, 4.2.10 we are
now ready to prove our main Theorem.
Proof of Theorem 4.2.1. Using a rather elementary argument, detailed in Secion 5 of
[FV15], it is possible to show that proving (4.38) is equivalent to showing that1
lim EHS(d ,d )= F$ (r).
x→∞(−q H(x,κx)−ηx+rγx1/3 ; q)∞
To do so we use formula (4.31) to express the q-Laplace transform on the left hand side.
We want to evaluatek+1
1 X (−1)k q ( 2 )limVx (ζq −k ) − Vx (ζq −k−1 ) ,x→∞ (q; q)∞(q; q)kk≥0(4.73)and to do so we aim to bring the limit inside the summation symbol. We start by fixing
a small number  and we split the summation in (4.73) into two different contributions.
One comes from the sum over k ranging in the region [0, γx1/3 ] and the other is given by
k in (γx1/3 , ∞). For each of these terms we can use different estimates.
We start with the latter, that is we take k > γx1/3 . A general inequality that can be
deduced from the definition of Vx = limv→d Vx;v,d and from Theorem 4.1.3, in case ζ is anegative number, is


1
1
−k
EHS(v,d )⊗M
Vx (ζq ) = limv↑d 1 − v/d(ζq −k q H−M ; q)∞


1
1
∗
≤ lim
EHS(v,d )⊗M= Vx (ζq −k ),∗ H−M−kv↑d 1 − v/d
(ζq q
; q)∞
which holds for every k ∗ < k. By taking k ∗ = γx1/3 and r∗ = r − we obtain the estimate


k+1
k+1
∗
1/3
q ( 2 ) Vx (ζq −k ) − Vx (ζq −k−1 ) ≤ 2q ( 2 ) Vx (ζq (r −r)γx )k+1
2= 2q ()

γx1/3 χ$ (r∗ ) +

4
X
i=1

114

!

S (i) (r∗ )Rx(i) (r∗ ) + O(x−1/3 ) .
(4.74)

In the right hand side of (4.74) we used results of Propositions 4.2.7, 4.2.8, 4.2.9, 4.2.10 toprovide the approximate expression of Vx . Function χ$ was defined in (4.40) and terms
S (i) ’s are explicit, bounded functions which for convenience we do not report explicitly.
We can therefore write
X (−1)k q (k+1
2 )

1
2/3
Vx (ζq −k ) − Vx (ζq −k−1 ) = o(e−cx ),(4.75)(q; q)∞(q; q)k1/3
k>γx

for some positive constant c, since, from (4.74) we see that the right hand side is a quantity
k+1
exponentially small in x2/3 , due to the presence of the term q ( 2 ) .
We now consider the contribution of the summation in (4.73), when the index k issmaller than γx1/3 . Once again, using results of Propositions 4.2.7, 4.2.8, 4.2.9, 4.2.10
we have


k
k+1
−k
−k−1
1/3
χ$ (r − 1/3 ) − χ$ (r − 1/3 )Vx (ζq ) − Vx (ζq) = γx
γx
γx


4
X
k
k
k+1
(i)
(i)
(i)
+
S (r − 1/3 ) Rx (r − 1/3 ) − Rx (r − 1/3 )
γx
γx
γx
i=1
+ O(x−1/3 ),(4.76)

which immediately implies
Vx (ζq −k ) − Vx (ζq −k−1 ) ≤ const,(4.77)

after expanding χ$ around r − γxk1/3 .

We can finally evaluate the limit (4.73). Using the bound (4.75), we write
 1/3

k+1
γx
X
X
(−1)k q ( 2 )(Vx (ζq −k ) − Vx (ζq −k−1 ))(4.73) = lim
+
x→∞
(q;
q)
(q;
q)
∞
k
1/3
k=0
k>γx

= lim

x→∞

X (−1)k q (k+12 )(q; q)∞ (q; q)kk≥0(4.78)(Vx (ζq −k ) − Vx (ζq −k−1 ))1[0,γx1/3 ] (k)and following estimate (4.77), we can employ the bounded convergence theorem to exchange the limit and summation symbols in the right hand side of (4.78). Here thepointwise convergence∂lim Vx (ζq −k ) − Vx (ζq −k−1 ) =χ$ (r)x→∞∂r
can be established through the expansion (4.76), using the fact that the difference between(i)remainder terms Rx ’s converges to zero, as reported in Propositions 4.2.7, 4.2.8, 4.2.9,
4.2.10. We can therefore write
X (−1)k q (k+1
2 )∂∂(4.73) =χ$ (r) =χ$ (r),(q; q)∞ (q; q)k ∂r∂r
k≥0

which concludes the proof.

115

4.2.3

Scaling form of determinantal formulas

In this Section we present expressions of kernels f (n), A(n, m), Φ(i) (n), Ψ(j) (n), %(n, m)
one finds in the definition of function Vx given in (4.32), that are amaenable to rigorous
asymptotic analysis.
We fix a parametrization of integer indeces n, m asn = nν = −ηx + νγx1/3 ,m = mθ = −ηx + θγx1/3 .

(4.79)

Here ν, θ belong to the set of rescaled integers
e = {ν ∈ R| − ηx + νγx1/3 ∈ Z}
Z

Pe
and we will use the symbol f to denote a summation where the index ranges over Z
rather than Z. We also introduce the scaling function
g(z) = −η log(z) + κa−1 (z) − h−1 (z),(4.80)wherexa−1 (z) = log(1 − zu)

and

1X
log
h−1 (z) =x y=2

(zsy /ξy ; q)∞(z/(ξy sy ); q)∞.

Functions ak , hk , for k ≥ 0, were defined in (4.34) and they satisfy the propertieszdak (z) = ak+1 (z)
dz

and

z

d
hk (z) = hk+1 (z),dz

for all k ≥ −1.
The combination of the KPZ scaling (4.58) and of the change of variable (4.79) issummarized in the following:
Proposition 4.2.11. Assume (4.58), (4.79) and fix a real number L such that

Define the sequence− L < r.

(4.81)(
1/3
ς νγx , if ν > −L,
τe(ν) =
τ (nν ), if ν ≤ −L,(4.82)

where ς is a number in a neighborhood of order x−1/3 of d and its exact expression in
given below in (4.86). Define alsoe θ) = τe(ν)
A(ν,
τe(θ)

Z

fe(ν) = f (nν ) =dw
D 2πiwZ

1
1 + q (ν−r)γx1/3,(4.83)

1/3

dz z θγx exg(z) (qd /w, qz/d ; q)∞ 1,
νγx1/3 exg(w) (qd /z, qw/d ; q) z − w∞
C 2πi w
116

(4.84)

−νγx1/3 −1 −xg(d )
e (1)
Φ
e
,
x (ν) = dνγx1/3 xg(d )
e (1)
Ψ
e
,
x (ν) = dZdw −νγx1/3 −xg(w) (qd /w; q)∞ 1we,(qw/d ; q)∞ w − d
D1 2πiwZ
dz νγx1/3 xg(z) (qz/d ; q)∞ 1
(2)
e
z
e
.
Ψx (ν) =(qd /z; q)∞ z − dC1 2πie (2)Φx (ν) =e x (ν) = Ψe (1) (ν) + Ψ
e (2) (ν)
Ψ
x
x

e (2)
e x (ν) = Φe (1)Φx (ν) + Φx (ν),

Then formula (4.31) for the q-Laplace transform still holds if we substitute, in the ex(i)
(j)
eΦ
e (i)
e (j) e epression of Vx (4.32), f, A, Φx , Ψx , Φx , Ψx with fe, A,
x , Ψx , Φx , Ψx and we change thePPsummation signs
with f.

Proof. We can easily see that the tilde notation corresponds to applying to functions in(4.32) the change of variables (4.79). The Fredholm determinant det(1−f A)l2 (Z) is clearlyτ (mθ )
not affected by the multiplication of A with the gauge factor τeτe(ν), nor by the change(θ) τ (nν )e 2 e . Similar considerations are trueof variables and it is therefore equal to det(1 − feA)l (Z)also for the remaining functions in (4.32).

The function g was used to simplify the expression of integrands of quantities A, Φ(i) , Ψ(j) .
Its crucial feature is that, in a neighborhood of size x−1/3 of d , it admits the expansion
g(z) = g(ς) + g 000 (ς)(z − ς)3+ o(x−1/3 ),6!

where ς is another point in a neighborhood of size x−1/3 of d . In other words g has adouble critical point ς in the vicinity of d and this will enable us to analyze the asymptotic
form of A, Φ(i) , Ψ(j) through saddle point method. This is precisely stated in the following:
Proposition 4.2.12. With the choice $ = 0, the function g has a double critical pointat d , that is g 0 (d ) = g 00 (d ) = 0. When $ 6= 0, there exists a point ς = ς($) such thatγ3g 000 (ς) = −2 3 + O(x1/3 )ς

g 0 (ς) = g 00 (ς) = O(1/x),(4.85)and g 000 (ς) < 0 for x large enough. Moreover, such ς admits the expansion



$
1 $2
a21 h3 − a1 a2 h2 + 2a22 h1 − a1 a3 h1
1++ o(1/x2/3 ),ς = d 1 + 1/3 +γx
2 γ 2 x2/3
a1 (a2 h1 − a1 h2 )(4.86)
where ak = ak (d ), hk = hk (d ) are as in (4.34).
Proof. Equalities reported in (4.85) can be verified by direct inspection making use of theapproximate form of ς (4.86). Therefore the only thing we are left to prove is that γ is apositive quantity. From expression (4.35) we write1h1 (d )3γ =h2 (d ) −a2 (d ) .
(4.87)2
a1 (d )
117

Functions a1 , a2 have the explicit expressions
J−1
X−d uq ja1 (d ) =,(1 − d uq j )2
j=0

a2 (d ) =J−1
X−d uq j (1 + d uq j )j=0(1 − d uq j )3,that can be recovered using the form (A.12) to compute υ1 , υ2 . On the other hand,
expressing the υk ’s in h1 , h2 using (A.11) we can write, after some algebraic manipulations,the right hand side of (4.87) as

k


x
J−1
X
k(1 − s2k
1 1 XXd−ud q j1 + ud q jy )k−,
2 x y=2 k≥1 ξy sy
1 − q k j=0 (1 − ud q j )2
1 − ud q jthat is a sum of positive terms since u < 0.

4.2.4

Proof of Proposition 4.2.7

In this Section we present the proof of Proposition 4.2.7 that establishes the convergenceof the Fredholm determinant det(1 − f A) to the GUE Tracy-Widom distribution. Rather
than using the original expressions (4.6) and (4.14) for f and A we will use their riscaled
e discussed in Proposition 4.2.11. Before we move to the more rigorous part offorms fe, A

118

the presentation we like to outline the strategy we will follow:
e θ)
Step 1 We first establish the pointwise convergence of the kernel A(ν,to the Airy kernel KAiry (ν, θ) and this is done in Lemma 4.2.13. Our
analysis relies on a saddle point method and because of this we cancontrol the error term up to order x−1/3 for (ν, θ) lying in an enlarging
rectangle [−L, xδ/3 ]2 with δ ∈ (0, 1/3). This corresponds to the redregion in Figure 4.1. The reason why we consider the convergence onrectangles that grow with x is that this will allow us to control thee and F2 as a function of x (see also
difference between det(1 − feA)
Remark 4.2.18).

e θ) for (ν, θ) in the set
Step 2 We estimate the decay of the kernel A(ν,2δ/3 2[−L, ∞] \ [−L, x ] , corresponding to the yellow region in Figure4.1. This is also done through a saddle point analysis; see Lemma
4.2.14.

(4.88)

Step 3 Combining the exponential decay in x of fe(ν) for ν < r and the facte θ) are bounded in modulus by 1 (see Lemma 4.2.15),
that terms A(ν,e 2 e with det(1 − feA)e 2e
we can estimate det(1 − feA)l (Z)l (Z≥−L ) up to anerror of order e−const.x

1/3. This is the result of Lemma 4.2.16.

Step 4 As a result of the exponential decay of front tails of the kernele θ) we can further approximate det(1 − feA)
e 2e
A(ν,
l (Z≥−L ) with det(1 −1/3−const.xe 2efeA). This is the resultl (Z∩[−L,xδ/3 )) up to an error of order eof Lemma 4.2.17.

e 2eStep 5 We can finally evaluate the convergence of det(1 − feA)l (Z∩[−L,xδ/3 ))to the Tracy-Widom distribution and we can verify the ”continuity”(1)
properties of the remainder Rx (r∗ ) in a neighborhood of r.
Let us now start.
Step 1: we have the following Lemma.
Lemma 4.2.13 (Convergence on moderately large sets). Let δ be a number in the interval(0, 1/3). Then for (ν, θ) ∈ [−L, xδ/3 ]2 we have 2 ,e θ) =A(ν,112δ/3−1
K
(ν,
θ)
+
Q(ν,
θ)
+
O
x
Airy
γx1/3
γ 2 x2/3(4.89)and the error term satisfies

x2/3 O(x2δ/3−1 ) −−−→ 0,x→∞(4.90)

uniformly in the sequence of sets (ν, θ) ∈ [−L, xδ/3 ]2 . Moreover the exponential estimates,|KAiry (ν, θ)|, |Q(ν, θ)| < c1 e−c2 (ν+θ) ,2

for motivation on the choice of sets [−L, xδ/3 ] see Remark 4.2.18.

119(4.91)

θ

xδ/3

−L

xδ/3

ν

−L

e θ) in the red the region (see Lemma 4.2.13) and
Figure 4.1: We estimate the kernel A(ν,in the yellow region (see Lemma 4.2.14). Because fe(ν) converges to the indicator function
1(r,∞) and −L < r, the contribution to the Fredholm determinant of feAe of integrationsin the blue region is negligible (see Lemma 4.2.16).
hold for all (ν, θ) ∈ [−L, xδ/3 ]2 , for an opportune choice of positive constants c1 , c2 which
do not depend on x.
Proof. The definition itself of scaling parameters is functional to perform a saddle point
analysis. In particular we want to show that, when ν and θ are relatively small quantities,
compared to x1/3 , the integrals in (4.84) are dominated by the value of the integrands atthe double critical point ς. To do so we suitably deform contours C, D in such a way that,for x large enough, the following properties hold:3
1. maxz∈C Re{g(z)} = g(ς(1 − 2γx11/3 ));2. maxz∈C |z| = ς(1 − 2γx11/3 );3. minw∈D Re{g(z)} = g(ς(1 + 2γx11/3 ));4. minw∈D |w| = ς(1 + 2γx11/3 ).
The idea is to take paths like those depicted in Figure 4.2. Based on results of Appendix
C.2, we now construct the steep descent contour C. The same procedure can be applied toprovide an exact expression for D as well and therefore we will omit this in the discussion.
3

for the sake of the uniform convergence over compact sets conditions 2,4 are not necessary, but westill state them as they will become useful later in Lemma 4.102.

120

Fix an arbitrarily small positive number  and consider C to be the union of two curvesee2 such that
C1 , Con√e1 = ∂D (0, ς(1 − )) ∩ z ∈ C| Re(z) ≤ ς (3 + 1 − 8 + 42 ) ,(4.92)
C
4(!
√
3
4 + γ 2 x2/3 ρ2e
C2 = 1[0, 2 ] (|ρ|)ς 1 −
ρ
+i
1/3
γx1/3
8γx
2
)
√ !
√
|ρ|
3
1
+1[ 2 ,∞) (|ρ|)ς 1 −+i
ρ : |ρ| ≤ (1 − 1 − 8 + 42 ) ,(4.93)
γx1/3
2
22
where ∂D(c, R) indicates a circumference of center c and radius R. To put it in simple
terms C is a circle of radius ς(1 − ) up until it intersects for the first time (from the left)(as in Figure 4.2, b)). After C meetsthe two complex lines exiting from ς with slope ± 2π3e2 , a regular curve which coincides
these intersection points, denoted with p± , it becomes Cwith such lines for a while and passes strictly to the left of ς.
We claim that the contribution of the integral in the z variable in (4.84) are given, up
e2 . Toto an error which is exponentially small in x, by the integral along the contour Cshow this, we first notice that from Proposition C.2.1, if  is small enough we can assume
e1 the real part of g(z) is a decreasing function. Therefore, the contribution
that, along C
e1 ,of the term exg(z) can be estimated by its values at the extremal points of C
√


√ 
√
√
ς
3
2
3 + 1 − 8 + 4 ± i
ς(1 − 1 − 8 + 42 ) ≈ ς 1 −  ± i 3 .
p± =
4
4

Let us evaluate the quantity Re{g(p± )} − g(ς) through a Taylor expansion. By using(4.85), we have
8g 000 (ς)ς 3 3
Re{g(p± )} − g(ς) = + R()4 ,3!
where R() is the Taylor remainder and it is a regular, bounded function in a neighborhood of zero. The factor ς 3 g 000 (ς) is strictly negative, as stated in Proposition 4.2.12 andtherefore we obtain the boundex(g(z)−g(ς)) ≤ e−cx ,e1 ,for each z ∈ Cwhich holds for some positive constant c.
Through an analogous argument we can deform the D contour too and separate it ine 1 and D
e 2 (see Figure 4.2). As for the C contour case, we canan union of two curves D
e 2 to be a curve that follows the two complex half lines {ς + e±i π3 ρ : ρ ≥ 0} in atake Dneighborhood of size  of ς and that passes strictly to the right of ς. For  small enough,e 1 can be chosen so that the contribution of
but still of order 1, the remaining contour D
e 1 to the kernel A
e are exponentially small in x.
the w integral over Dee
We also remark that curves C2 , D2 are kept at a distance of size x−1/3 from ς (ande of a singularity
hence from each other) due to the presence in the integral expression of A
at z = w.
121

b)

a)

p+

e1
C

e1
D

D
e2
C

C

ς

e2
D

ς
ς
2γx1/3p−ςFigure 4.2: a) Choices of integration contours in Lemma 4.2.13. The red contour C encircles thesigularities {q k d }k≥1 , is contained inside a circle of radius ς(1−) and joins the point ς(1− 2γx11/3 )with slope π3 from above (resp. − π3 from below). The green contour D contains the singularities{ξk sk }k=2,...,x and forms at the point ς(1 + 2γx11/3 ) a cusp of width 23 π, symmetric to that of C.
b) A representation of contours C and D in the immediate vicinity of the critical point ς.

e as
We can summarize discussion made so far expressing the kernel A
1/3

νγx
1
e θ) = ςA(ν,1/3ς θγx (2πi)2

Z

Z

1/3

z θγx exp{xg(z)} (qd /w, qz/d ; q)∞ 1+O(e−cx ),dz νγx1/3
exp{xg(w)}
(qd
/z,
qw/d
;
q)
z
−
w
e
w
∞
C2
(4.94)
e2
where we notice that, with respect to (4.84), the integration contours have become Ce 2 and the remainder is a quantity which decays as an exponential in x. We canand D
now safely employ the saddle point method to give an estimate of the integral expressionin (4.94). The only significant contribution to the double integral (4.84) is given when
variables z, w are separated from ς by a distance of order x−1/3 . For this reason we liketo apply the change of variables




Z
W
z = ς 1 − 1/3 ,w = ς 1 − 1/3 ,
γx
γx
dw
e2 w
D

and we write, through simple Taylor expansions, different terms of the integrand function122

in (4.94) as



 3

1/3
1/3
ς θγx e−θZz θγx1
νW 2 θZ 2
Z θ W 3 ν Z 4 θ2 W 4 ν 2= νγx1/3 −νW 1 + 1/3
−
+O
,
,
,
,
e
γx
2
2
x2/3 x2/3 x2/3 x2/3wνγx1/3
ς(4.95)
3
Z8 W 8exg(z)
eZ /3
1
4
4
+
O
E
Z
−
E
W
,
(4.96)
=
1
+
,
1
1
3
exg(w)
γx1/3
x2/3 x2/3
eW /3

 2
(qd /w, qz/d ; q)∞1
Z W2= 1 + 1/3 (E2 Z − E2 W ) + O.
(4.97),(qd /z, qw/d ; q)∞γxx2/3 x2/3
In these expressions, coefficients E1 , E2 , naturally possess exact expressions, which we do
not report as they are irrelevant for the computations.
e in the infinitesimalThanks to (4.94),(4.95),(4.96),(4.97) we obtain an expansion of A
1/3
1/3
quantity 1/(γx ). Collecting together terms of order 1/(γx ) and 1/(γx1/3 )2 we obtainZ e 23 πi ∞

dW
πi
−2
e 3 ∞ 2πie θ) = 1
A(ν,
γx1/3

Z e− π3 i ∞πe 3 i∞

3

1dZ eZ /3−θZ
1+
Q(ν, θ) + O(x2δ/3−1 ).
3 /3−νW
1/3
2
W
2πi e
W − Z (γx )(4.98)

with the kernel Q being given byZ e 32 πi ∞Z e− π3 i ∞3νW 2 θZ 2−+ E1 (Z 4 − W 4 ) + E2 (Z − W )
πi
2
2
e3 ∞
(4.99)e as in (4.89).
By recognizing the expression of the Airy kernel (4.39) in (4.98) we write A
All we are left to do is to prove the exponential bound (4.91) for Q, since the same
type of estimate for KAiry follows from well known decay properties of the Airy functions.
To do this consider the following parametrization of the integration variablesdW
Q(ν, θ) =2
e− 3 πi ∞ 2πidZ eZ /3−θZ
2πi eW 3 /3−νWebeb+ |%1 |e− sign(%1 )iπ/3 ,W = − + |%2 |e− sign(%2 )i2π/3 ,(4.100)2
2
for ρ1 , ρ2 ∈ R and eb being a positive real number. Applying the substitution (4.100) in(4.99), we straightforwardly obtain an inequality likeZ=eb3 /12 Z ∞e− 2b (θ+ν) e|Q(ν, θ)| < e

eb

0

d%1

Z ∞
0




d%2 |θ|Peb (ρ1 ) + |ν|Peb (ρ2 ) + Seb (%1 ) + Seb (%2 )




e2
e2
e
− b (ρ21 +ρ22 )− θ2 − b8 ρ1 − ν2 − b8 ρ2

×e 4

,

(4.101)

where Peb and Seb are polynomials and by making use of elementary estimates on the
integrals on the right hand side of (4.101), we can finally show (4.91).
The error term O(x2δ/3−1 ) in (4.89) is obtained taking into account quantities

 8

 2

 3
Z W8
Z W2
Z θ W 3 ν Z 4 θ2 W 4 ν 2
,
,
,
,O
,
,O
,
, O(e−cx )Ox2/3 x2/3 x2/3 x2/3
x2/3 x2/3
x2/3 x2/3
123



1
.
W −Z

from (4.94) and (4.95), (4.96), (4.97) in the saddle point integration. Due to the presence3
3of the exponentially decaying term eZ /3−W /3−Zθ+W ν we can formulate bounds like (4.101)
for these remainders as well, to finally show (4.90). This concludes our proof.
e θ) in the yellow
Step 2: the following lemma establishes the exponential decay of A(ν,region in Figure 4.1.

Lemma 4.2.14 (Exponential decay of front tails). Let L0 be an arbitrary large positive
real numbers (possibly of order x raised to some power). Then there exists x∗ , such thatfor all x > x∗ the bounde θ) < e−ν−θ
γx1/3 A(ν,(4.102)holds for each (ν, θ) ∈ [−L, ∞)2 \ [−L, L0 ]2 .

Proof. We use again suitable deformations of contours described in Lemma 4.2.13 to
estimate, for large x, the contribution of the factor
1/3

z θγx
wνγx1/3
to the double integral (4.84). Let’s first prove (4.102) in the case θ ≥ ν. When this ise1 ∪ Ce2 ,the case, we take the contour D exactly as in Lemma 4.2.13 and we modify C = C
where


2ς
3
e1 = ∂D 0, ς −
)},
C
∩
{z
∈
C|
Re(z)
≤
ς(1
−
γx1/3
γx1/3
e2 = ς(1 − 3 ) + i[−ςe
a, ςe
a]
C
γx1/3

and e
a is given by the intersections of the vertical complex line {ς(1 − γx31/3 ) + iy| y ∈ R}
2ςwith the circle ∂D 0, ς − γx1/3 . We can also write down its exact expression as
r

r
5
2
2 1
e
a=
−
≈
+ O(x−1/3 ).
1/3
2
2/3
1/6
γx
γ x
γx


From Proposition C.2.1, ∂D 0, ς(1 − γx21/3 ) is a steep descent contour for Re(g) and we
can assume that3
max Re{g(z)} = Re{g(ς(1 − 1/3 + ie
a))}.
z∈C1
γx

e2 we use theTo evaluate the real part of the function g on the complex segment C
parametrization3Zz = ς 1 − 1/3 + i 1/3 .
(4.103)
γx
γx

124

C̃1
ς(1 − γx31/3 + iã)
π/3
ς
D

C̃2
2ς
γx1/3

3ς
γx1/3

Figure 4.3: Choice of integration contours of Lemma 4.2.14. The red contour C is the union ofe1 , an arc of the circle of center 0 and radius ς(1 − 21/3 ), and C
e2 , a vertical segment passingC
γx

for the point ς(1 − γx31/3 ) on the real axis. On the other hand D, in the vicinity of the criticalpoint ς, stays close to the lines exiting from ς with slope ± π3 (dotted lines).

In this case√Z is a real
ranging in an interval which, up to corrections of order√ number
1/6
1/6
−1/3
x
is [− 2γx , 2γx ]. Expanding g in Taylor series around ς and recalling (4.85),
we have 2
1 ς 3 g 000 (ς)
1 ς 4 g (4) (ς) 4
Z
Z4 Z6
2
Re{g(z)}−g(ς) =(−27+9Z )+ 4/3
Z +O,,, (4.104)
x 3!γ 3x4!γ 4
x4/3 x5/3 x2
where the presence of terms of order higher than three takes into account the fact that Z
can be of order x1/6 . When Z/(γx1/3 ) = ea, (4.104) becomes1
1 4 (4)33 000a))} − g(ς) = 2 2/3 3ς g (ς) + ς g (ς) + O(x−1 )Re{g(ς(1 − 1/3 + ie
γxγ x
6and the term on the right hand side of order x−2/3 is negative. This can be shown either
directly computing the derivatives of g or simply recalling that the point ς(1 − γx31/3 + iea)lies on a steep descent contour. These calculations imply the estimate|ex(g(z)−g(ς)) | ≤ e−cx

1/3

,

e1 ,
for each z ∈ C(4.105)e2 , (4.104) gives us
for some positive constant c. On the other hand, when z belongs to Cthat2
|ex(g(z)−g(ς)) | ≤ e9−ecZ ,for each |Z| ≤ γe
ax1/3 ,
(4.106)for some other positive constant ec.
To complete the list of preliminary estimates for terms depending on z in the integral
e we need to address the factor z θγx1/3 . First we notice that,
formula (4.84) of the kernel A,
since the contour C lies inside the circle centered at 0 with radius ς(1 − 2/(γx1/3 )), we
125

have
z
ς

θγx1/3
1/3≤ exp θγx log 1 −

2
γx1/3



≤ e−2θ ,for each z ∈ C,

(4.107)as a result of the simple inequlity log(1 + y) ≤ y, valid for all y > −1. Moreover, when z
e2 , using the parametrization (4.103), we haveis on Czςθγx1/3

1/3= exp θγx3iZ
log 1 − 1/3 + 1/3
γx
γx







9 + Z2≤ exp −θ 3 −. (4.108)2γx1/3e we also need to provide some estimates for quantities involving
To evaluate the kernel A
the variable w. The choice of contours C, D implies that
1
ς
≤ 1/3
z−w
γx

(qd /w, qz/d ; q)∞≤ Γ1 ,(qd /z, qw/d ; q)∞and(4.109)for some constant Γ1 . In addition, since ν > −L and |w| > ς, combined with the fact that
D is steep ascent for the function Re{g}, as proved in Proposition C.2.3, we have thatς −Lγx1/3
ς νγx1/3
exp{x(g(ς) − g(w))} ≤exp{x(g(ς) − g(w))} ≤ Γ2 ,w
w

(4.110)for some other constant Γ2 . Combining together inequalities (4.108), (4.109), (4.110), we
can write
Z
1/3
1/3 Z
dw z θγx exp{xg(z)} (qd /w, qz/d ; q)∞ 1
ς νγx
dz
e
|A(ν, θ)| = θγx1/3
νγx1/3 exp{xg(w)} (qd /z, qw/d ; q) z − wς∞C 2π D 2πw w(4.111) θγx1/3
Z
Γ1 Γ2 l(D)z≤dz
exp{x(g(z) − g(ς))} ,(2π)2 γx1/3 Cςwhere l(D) is the length of the curve D. The integral over C is naturally split into different
e1 and C
e2 . On C
e1 , utilizing (4.105) and (4.107) we
contributions coming from contours Chave θγx1/3
Z
z
−1/3
e1 ),
dz
exp{x(g(z) − g(ς))} ≤ e−2θ e−cx l(C(4.112)ςeC1e2 , from (4.106), (4.108) we obtain
whereas on C θγx1/3
 
Z aγx1/3z
9 + Z22exp{x(g(z) − g(ς))} ≤dZ exp −θ 3 −
+9−e
cZ .
dz
ς
2γx1/3e2
C−aγx1/3(4.113)
To estimate the integral on the right hand side of (4.113), set a large integer N andsplit the integration segmant into |Z| < N and N < |Z| < eaγx1/3 . When |Z| < N the9+Z 2
term 2γx1/3 is small and we can denote it with O(N 2 /x1/3 ). On the other hand, when
Z

126

2

9+Z
N < |Z| < eaγx1/3 , since (3 − 2γx1/3 ) > 2, the integrand becomes very small due to thepresence of the exponential of −ecZ 2 . We can therefore write


Z
Z N
9−e
cZ 2
9−e
cZ 2
−θ(1−O(N 2 /x1/3 ))
−2θ
e
dZ +
dZe
e
rhs (4.113) ≤ e−N
N <|Z|eaγx1/3(4.114)−2θ−θ(1−O(N 2 /x1/3 ))
−e
cN 2
=e
e
Γ3 + O(e) ,with Γ3 being a constant coming from the integration of the exponential.
We can now plug (4.112), (4.113), (4.114) into the right hand side of (4.111) to finallyobtaine θ)| < e−2θ e−cx1/3 l(C1 ) + e−θ(1−O(N 2 /x1/3 )) Γ3 + O(e−ecN 2 ) Γ1 Γ2 l(D) .
|A(ν,(4.115)(2π)2 γx1/3
2

taking x  0 and
The term inside the parentheses can be made smaller than Γ1(2π)Γ2 l(D)00
L  0 (remember L < θ), so that (4.115) reduces toe θ)| < e−2θ
|A(ν,

1
,
γx1/3

which implies (4.102) since −2θ < −θ − ν.
The complementary case ν > θ can be studied analogously, deforming the contour D,
instead of C, symmetrically with respect to the critical point ς.
e in a region where both θ and ν are bounded
Up to this point we estimated the kernel A
from below. When this is not the case the saddle point method cannot be applied any
longer as the contribution to the integral (4.84) of the term
1/3

z γx θ
wγx1/3 νeis no more negligible. In the following Lemma we show how to control the rear tails of A.

e with kAk
e defines a trace class operator on l2 (Z)e = 1. In
Lemma 4.2.15. The kernel A
particular we have the bounde θ)| ≤ 1.
|A(ν,(4.116)e is obtained from the kernel A through a simple change of variable and a
Proof. Since Aτ (mθ )
, we can still write the expansion
multiplication by a gauge factor τeτe(ν)(θ) τ (nν )e θ) =
A(ν,

x−1
X
τe(ν)

x−1
X
τ (mθ )
φl (nν )ψl (mθ ) =φel (ν)ψel (θ),
τ
(n
)
τ
e
(θ)
ν
l=1
l=1

θ)
where φel (ν) = ττe(n(ν)ν ) φl (nν ) and ψel (θ) = ττe(mψl (mθ ). Functions φel , ψel , like φl , ψl , are(θ)
still a biorthogonal family and to prove this we only have to check that the summation

127

P
f φel (ν)ψek (ν) is absolutely convergent. This can be done establishing exponential decay
νof tails of φel , ψek . For the rear tails this is done as in Appendix C.1 for functions φl , ψl(for the real tails), while front tails are estimated using a saddle point analysis as thate in Lemma 4.2.13, 4.2.14. Such exponential bounds imply that
performed for the kernel A
e is trace class for each x, following the argument in the proof of Proposition 4.1.4. Also,Aethe biorthogonality of φel , ψek implies that kAk=
1 as shown in the proof of Proposition
4.1.6.
e
Step 3: we can now start the evaluation of the Fredholm determinant of the kernel feA.
Lemma 4.2.16. There exist constants c, x∗ such that, for each x > x∗ , we havee
det 1 − feAel2 (Z)

e− det 1 − feA

1/3el2 (Z∩[−L,∞))

Proof. First we set constants c1 , c2 such that the estimate(q1c e−θ−ν ,e θ) < γx1/3 1
fe(ν)fe(θ)A(ν,1/3
1
ec2 (min(θ,−L)+min(ν,−L))γx ,γx1/3< e−cLγx.

(4.117)if θ, ν ∈ [−L, ∞),else,

holds, for each x sufficiently large. This is always possible as a result of Lemmas 4.2.13,
4.2.14, 4.2.15 and from the fact that fe(ν), given in (4.83), decays exponentially in x1/3
when ν  0. In particular we can easily deduce the additional boundqe θ) < c3 1 e−|θ|−|ν| ,fe(ν)fe(θ)A(ν,γx1/3
true for any θ, ν, for some constant c3 . We havelhs of (4.117) =X (−1)kk≥1≤

k!

X (−1)kk≥1Xgk(ν1 ,...,νk )∈[−L,∞)
/

X
g

X

(k − 1)! ν ≤−L ν ,...,ν
1

2

q

e
e
e
det
f (νi )f (νj )A(νi , νj )
k

i,j=1

q

e
e
e
f (νi )f (νj )A(νi , νj ) .
det(4.118)

k

i,j=1
k

Thanks to the Hadamard’s inequality we can estimate the determinantal term in the sum
as
!1/2
q

k
k
Y
X
k
2
e i , νj ) ≤e i , νj )detfe(νi )fe(νj )A(νfe(νi )fe(νj ) A(ν
i,j=1

i=1

j=1

k
k k/2 c2 ν1 γx1/3 k−1 Y −|νj |
e
,
e
c3
≤
(γx1/3 )kj=2

so that, using this bound in (4.118), we obtain our result.
128

Step 4:
Lemma 4.2.17. Take constants L, δ such that −L < r and δ ∈ (0, 1/3). Then there exist
constants C, x∗ such that, for each x > x∗ , we havee
det 1 − feAe ≥−L )
l2 (Z



e
− det 1 − feAδ/3δ/3 ))el2 (Z∩[−L,x< Ce−x.

(4.119)Proof. The proof of (4.119) makes use of the exponential bound (4.102) choosing L0 = xδ/3
and it is similar to that of Lemma 4.2.16, therefore we omit it.
Step 5: we can finally combine all previous preliminary Lemmas and give the proof of
Proposition 4.2.7.
Proof of Proposition 4.2.7. To prove this result we first use Lemma 4.2.16 and Lemmae ∩ [−L, xδ/3 ]).
e in l2 (Z4.2.17 to restrict our attention to the Fredholm determinant of feA
The error we make while considering this restriction is exponentially small in x and hence
it is irrelevant when it comes to a decomposition like (4.61). Using results of Lemma
4.2.13 we havee θ) =A(ν,11KAiry (ν, θ) +Q(ν, θ) + O(x2δ/3−1 ),1/3
γx(γx1/3 )2fe(ν) = 1[r,∞) (ν) + ∆r (ν),

where the term ∆r is simply expressed as1 (ν−r)γx
if ν < r,
1/3 ,1+q∆r (ν) = −q(ν−r)γx1/3
, if ν ≥ r.
(ν−r)γx1/3
1+qe θ) based on their order in x−1/3 as
We can saparate the terms of the product fe(ν)A(ν,1
1[r,∞) (ν)Q(ν, θ),
γx1/3
1
B (4) (ν, θ) = 1/3 ∆r (ν)Q(ν, θ),γxB (1) (ν, θ) = 1[r,∞) (ν)KAiry (ν, θ),B (2) (ν, θ) =B (3) (ν, θ) = ∆r (ν)KAiry (ν, θ),(5)Bi,j (ν, θ) = O(x2(δ−1)/3 ).
e asIn this notation we write the Fredholm determinant of feAkX (−1)k
X
k

1
g
(1)
(5)
e
e
det
B
(ν
det 1 − f A=
1+
i , νj ) + · · · + B (νi , νj )1/3
δ/3 ])
e
i,j=1
k!
γx
l2 (Z∩[−L,xδ/3k≥1νl ∈[−L,x ]
l=1,...,k(4.120)and our goal is to separate the contribution of higher order terms B , . . . , B from thatof B (1) . To do so we use a formula that expresses the determinant of a sum of matrices
(2)

129

(5)

B (1) + · · · + B (N ) in terms of sums of determinants of matrices having for column i, thei-th column of exactly one of the B (1) , . . . , B (N ) . More precisely we haveX
det B (1) + · · · + B (N ) =det B (I1 ,...,IN ) ,(4.121)∪N
i=1 Ii ={1,...,k}Ii ∩Ij =∅ if i6=j

where
(I ,...,IN )

Bi,j1

(l)

= Bi,jif i ∈ Il .

This expansion holds for generic k × k matrices B (1) , . . . , B (N ) and it follows directly fromthe multi linearity of the determinant. Using (4.121) can rewrite the determinant in theright hand side of (4.120) asX(4.122)det B (1) + · · · + B (5) = det B (1) +det B (I1 ,I2 ,I3 ,I4 ,I5 ) .
∪5i=1 Ii ={1,...,k}Ii ∩Ij =∅ if i6=j
I1 6=∅The Hadamard inequality provides the bounddet(B (I1 ,I2 ,I3 ,I4 ,I5 ) ) ≤

5 Y X
k
Y
l=1 i∈Il

j=1|B (l) (νi , νj )|2

!1/2

,

while exponential inequality (4.91) allows us to write
B (1) (νi , νj ) < c1 e−c2 νi ,B (2) (νi , νj ) <1
c1 e−c2 νi ,
γx1/3

1/3

B (3) (νi , νj ) < c1 q |νi −r|γx1/3

B (4) (νi , νj ) <c1 q |νi −r|γx
γx1/3

,

B (5) (νi , νj ) = O(x2(δ−1)/3 ).

Integrating the generic term of the summation in (4.122) we obtainXgνl ∈[−L,xδ/3 ]
l=1,...,k



1
γx1/3

k


det(B (I1 ,I2 ,I3 ,I4 ,I5 ) ) = k k/2 O x−|I2 |/3−|I3 |/3−2|I4 |/3−|I5 |(2/3−δ) ,(4.123)where the exponents −|I3 |/3 and −2|I4 |/3 appear due to the fact that the function
1/3
q |ν−r|γx is exponentially small in x outside of a neighborhood of size x−1/3 of r. Because
at least one among I2 , . . . , I5 is not empty, we have proven that



1
e
e
Sx(1) (r)det 1 − f A
=
det
1
−
1
[r,∞) KAiry l2 (Z∩[−L,xδ/3 ]) +
e
1/3
δ/3
2
e
γx
l (Z∩[−L,x ])1= det 1 − 1[r,∞) KAiry L2 (R) + 1/3 Sx(1) (r) + Sx(2) (r) .
γx(4.124)130,

In the right hand side the error S (1) comes from contribution to the Fredholm determinantPof matrices B (2) , . . . , B (5) , while S (2) comes from substituting discrete integrations fwith integral symbols. Both these quantities are explicit and clearly bounded due to the
exponential estimates (4.91) and this proves (4.61), (4.62).
The continuity of the remainder term (4.63) can be proven following the same strategy(1)(2)
used to prove its boundedness. In fact for any r∗ , Sx (r∗ ), Sx (r∗ ) are sums of determinants of kernels depending on r∗ and to evaluate the differences
Sx(i) (r∗ ) − Sx(i) (r∗ − 1/γx1/3 )we can first expand these kernels around r∗ and subsequently analyse the contributionsof terms of order zero and one in x−1/3 using an expansion of the form (4.122). We don’t
discuss these details any further.
Remark 4.2.18. The statement of Proposition 4.2.7 not only tells us thate −−−→ F2 (r),det(1 − feA)x→∞

but also it gives us an estimate of the error depending on x and this will be essential(1)
in the proof of Theorem 4.2.1. To measure such error term, namely γx11/3 Rx in (4.61),
e 2 ) with its truncated version defined only on
e on l2 (Zwe approximated the kernel feAe ∩ [−L, xδ/3 ])2 . The choice of the supremum of the segment [−L, xδ/3 ] is actually very(Z
relevant and possibly differentiate our analysis of the Fredholm determinant from that ofe only on compact
earlier works, such as [BFS08]. Had we considered the convergence of feAsets like [−L, L0 ], with L0 being some finite constant, we would have ended up, in Lemma
4.2.17 (replacing every xδ/3 with L0 ), with a bound like0

lhs of (4.119) < Ce−L .

(4.125)This clearly would have not been enough for our purposes, as the right hand side of (4.125)
has no dependence on x and in particular does not decay when x becomes infinite.

4.2.5

Proof of Propositions 4.2.8, 4.2.9, 4.2.10

In this Section we carry out proofs of Propositions 4.2.8, 4.2.9 and 4.2.10, that werestated in Section 4.2.2 and used in the proof of our main result Theorem 4.2.1. As in
Section 4.2.4, rather than the original expressions of f, A, Φ(i) , Ψ(j) we will make use ofeΦe (i) , Ψe (j) discussed in Proposition 4.2.11. First we present thetheir rescaled forms fe, A,
proof of Proposition 4.2.8 and the argument we follow traces that given in Lemma 5.12of [IS19].
Proof of Proposition 4.2.8. First we see that the termX  q n /ζ
q n+1
+2
υ0 (1/ζ) + 2 υ0 (q) =
n /ζ
1
−
q
1 − q n+1
n≥0131

plays no role in the limit as it is a bounded quantity in x for each fixed r.
Less trivial is to calculate the limiting form of υ0 (qζ), which is a summation likeX

q n+1−X
,
n+1−X
1
+
q
n≥0

for X being large. The kicker here is understanding that the main contribution to thesum is given by terms where m runs between 0 and 2 dXe 4 . Coupling the (k − 1)th andthe (2 dXe − k − 1)th addends and using the simple inequality
1−

1 − q2
1
1≤+≤ 1,
1 + q 2 + q k−X + q X−k+2
1 + q X−k 1 + q X−2dXe+k

we see thatX

1
= X + O(1).
1 + q X−n−1n≥0

We are interested in the case when X = ηx − γx1/3 r, so that, plugging this result into
(4.64) we are left to calculate

1
1/3
κxa
(d
)
−
xh
(d
)
−
ηx
+
rγx
0
0
x→∞ γx1/3
x2/3 0d g (d ),= r + lim
x→∞ γ
lim

which gives (4.64) and (4.65) after expanding g 0 around its critical point ς as1
ς2
$2
g 0 (d ) ≈ g 000 (ς)(d − ς)2 = 2 g 000 (ς) 2/3 + o(x−2/3 ).
2
2γ
x
(3)

This procedure also proves the boundedness of the remainder Rx due to the generalityof r.
Result (4.66) follows from expression (4.64). We haveX  q n /ζ
q n+1 ζ(2) ∗1/3 ∗Rx (r ) = −γx r −−+ (terms independent of r∗ ),
n /ζ
n+1 ζ
1
−
q
1
−
q
n≥0
1/3 r ∗

where ζ = −q −ηx+γx

(2)

(2)

. In this way the difference Rx (r∗ )−Rx (r∗ −1/(γx1/3 )) becomes
1/3 ∗

1/3 ∗

q ηx−γx r
q −ηx+γx r−1 ++,1 + q ηx−γx1/3 r∗ 1 + q −ηx+γx1/3 r∗which converges to zero exponentially as x goes to infinity.
4

here de is the ceiling function132

Next we present the proofs of Propositions 4.2.9, 4.2.10. Our approach follows a saddle
point analysis analogous to that showed in Section 4.2.4. The only difference between theargument we present next and that used for the evaluation of the Fredholm determinante consists in the proof of the exponential decay of rear tails of fe(ν)Φ
e (i)
e (j)
of feAx (ν)Ψx (ν).
e was implied by the bound (4.116)
In fact in Section 4.2.4 the decay of rear tails of feA
and by the convergence fe → 1(r,∞) . Below to obtain similar estimates we use more directcomputations and hypothesis (4.60).
Proof of Proposition 4.2.9. By making use of the saddle point method it is easy, at thisstage, to obtain a convergence result asXX(j)e (i) (ν)Ψ
e (j) (ν) −−−→
1[r,∞) (ν)Υ(i)(4.126)fe(ν)Φd−$ (ν)Υ$ (ν)
xxi,j=1,2(i,j)6=(1,1)

x→∞i,j=1,2(i,j)6=(1,1)

and to estimate the error term depending on x and r. This holds for ν in relatively large
sets of the form [−L, xδ/3 ] for some fixed L > 0 and δ ∈ (0, 1/3). Also, assuming a suitably
strong decay of tails of summands in the left hand side of (4.126), this easily leads to an
expansion of type (4.67).
e (2)
e (2)
Using suitable deformations of contours in the integral expressions of Φx , Ψx , suchas those seen in Lemma 4.2.14, one can also establish an exponential type decay for thefront tail (ν  0) of (4.126).
The exponential decay we have in the left hand side of (4.126), when ν goes to −∞is slightly different from what seen previously and in particular, here we make use of the2a< q −1 d stated in (4.60). We evaluate separately each one of the threehypothesis 1+σsummands in the left hand side of (4.67), when (i, j) is either equal to (1, 2), (2, 1) or(2, 2).
We start with the (i, j) = (1, 2) term. From expressions reported in Proposition 4.2.11,
we writeZ1dz  z γx1/3 ν x(g(z)−g(d )) (qz/d ; q)∞
(1)
(2)
e
e
e
d f (ν)Φx (ν)Ψx (ν) =e. (4.127)(q; q)∞
1 + q γx1/3 (ν−r) C1 2πiz d
We take C1 to be a circle of center in 0 and radius ς(1 − h/(γx1/3 )), where h is chosen sothat |z| < d for all z in C1 (e.g. take h > $). When x is large enough, such C1 is a steep
descent contour for Re{g}, as proven in Proposition C.2.1. This, along with the fact that
ς is a double critical point for g allows us to state the bound| exp{x(g(z) − g(d ))}| < const

for all z ∈ C1 .

Moreover, the choice of C1 also allows us to write

 




h
z γx1/3 νw
1/3= exp γx ν log 1 − 1/3 − log 1 − 1/3+ o(1)dγxγx≤ exp {ν($ − h) + o(1)} ,133(4.128)(4.129)yhaving used the simple logarithmic inequality 1+y≤ log(1 + y) ≤ y, valid for all y > −1.
Despite the right hand side of (4.129) is a quantity which diverges exponentially when
1/3
ν → −∞, its contribution is easily balanced by the term 1/(1 + q γx (ν−r) ) in (4.127),
1/3
which, for ν < −L, decays as q −(1−r/L)γx ν . Following (4.128), (4.129) we come to theestimatel(C1 ) (h−$)ν −(1−r/L)γx1/3 νeq|(4.127)| < const
2π
where in the right hand side the constant term also includes a trivial bound for the factor(qz/d ;q)∞. This is enough to show that for L large enough, we have(q;q)∞
1/3 ν|(4.127)| < c1 ec2 γx

for all ν < −L,,(4.130)where c1 and c2 are two suitably chosen positive constants.
We now want to establish a type of bound similar to (4.130) for the term (i, j) = (2, 1)of the left hand side of (4.67). Again, from (4.23), (4.24) we write γx1/3 ν(qd /w; q)∞ ddex(g(d )−g(w)).
1/3 (ν−r)γxw(qw/d ; q)∞ w − d1+q(4.131)
As a contour D1 we can simply take the contour D described in Proposition C.2.3. Since
we can always deform the integration contour in a neighborhood of size x−1/3 of ς, without
loss of generality, we assume that d lies strictly at the left of D1 . With this choice, weknow that D1 is a steep ascent contour for Re{g} and this, along with the fact that ς isdouble critical point for g implies the bound
1

e (2)
e (1)
d fe(ν)Φ
x (ν)Ψx (ν) =Z

dw
D1 2πiw| exp{x(g(d ) − g(w))}| < const

for all w ∈ D1 .

Another consequence of the choice of contour D1 is thatmax |w| ≤

w∈D1

2a
,
1+σ

as reported in (C.8). This immediately gives us the estimatedw

γx1/3 ν(1 + σ)d≤2a

γx1/3 νfor all w ∈ D1 ,since in this case ν is taken to be negative. In expression (4.131), the contribution of thefactor d /(w − d ) is bounded, in absolute value, by a quantity of order x1/3 and therefore
we come to writel(D1 ) 1/3 (1 + σ) −(1−r/L)|(4.131)| < const
x
q
d
2π
2a

γx1/3 ν.

(4.132)

When L is large enough, the assumption 2a/(1 + σ) < q −1 d of (4.60) guarantees thatthe right hand side of (4.132) is bounded by an exponential function in γx1/3 ν wheneverν < −L and this concludes our analysis of the rear tail of the term (i, j) = (2, 1).
134

To obtain the same type of result also for the case when (i, j) = (2, 2) one can reproduce, with minor adjustments, the same argument we used for (i, j) = (2, 1) and therefore
we omit details on this part.
We have, at this point proved a bound for the summands in expression (4.67) of theformXi,j=1,2(i,j)6=(1,1)

c2 γx1/3 ν
e (j)
e (i)
fe(ν)Φ
x (ν)Ψx (ν) < c1 e

for all ν < −L,

for suitably chosen positive constants c1 , c2 . This concludes our argument.
We conclude this Section presenting the proof of Proposition 4.2.10.
Proof of Proposition 4.2.10. First we expand the expression in the left hand side of (4.70)
as
X
d
g e e
e x (λ2 )Ψ
e x (ν).
f (ν)A(ν, λ1 )e%(λ1 , λ2 )fe(λ2 )Φ
(4.133)1/3γxeν,λ1 ,λ2 ∈Z

We can split the summation (4.133) as




X
X
d 
g
g
e(λ2 )Φ
e λ1 )ee x (λ2 )Ψ
e x (ν) .
 fe(ν)A(ν,
%
(λ
,
λ
)
f
+
1
2
γx1/3
δ/3 3
δ/3 3
(ν,λ1 ,λ2 )∈[−L,x]ν,λ1 ,λ2 ∈[−L,x/]Using estimates already encountered in the proofs of Proposition 4.2.7, 4.2.9, we knowthat the contribution of summation where indices do not belong to [−L, xδ/3 ]3 is exponentially small in some power of x. On the other hand, when all ν, λ1 , λ2 belong to [−L, xδ/3 ],we can safely employ the saddle point method to estimate the summand terms in ande This would ultimately
obtain their expansion in power of x−1/3 , as done in (4.89) for A.
lead to the convergence result (4.70) and to a verification of properties (4.71), (4.72) for
the remainder term. The procedure is analogous to what explained throughout the restof the section and therefore we do not describe its details any further.

135

Chapter 5
Spin Whittaker functions from q → 1
limitThis chapter presents results contained in sections 7,8 and 9 of[MP20] M. Mucciconi and L. Petrov. “Spin q-Whitaker polynomials and deformed quantumToda”. In: arXiv preprint (2020). arXiv:2003.14260 [math.PR]

5.1

Spin Whittaker functions

In this section we introduce new one-parameter deformations of the gln Whittaker functions [Jac67], [Kos78]. These deformations arise from our version of spin q-Whittakerpolynomials in a scaling limit as q → 1. The deformation parameter is denoted by S > 0.

5.1.1

Whittaker functions

Before proceeding with deformations of Whittaker functions, let us recall the usual glN
Whittaker functions. These functions play a central role in representation theory andintegrable systems [Kos80], [Eti99], [Giv97] as well as are related to several models of
random polymers [OCo12], [Cor+14], [OSZ14], [BC14].
The glN Whittaker functions ψλ1 ,...,λN (uN ) are indexed1 by N -tuples uN = (uN,1 , . . . , uN,N ) ∈N
R , depend on λ = (λ1 , . . . , λN ) ∈ CN , and may be defined through the recursion (following from the Givental integral representation [Giv97], cf. [Ger+06]):
ψλ1 ,...,λN (uN ) =

Z

RN −1

→N −1
ψλ1 ,...,λN −1 (uN −1 ) QN
(uN , uN −1 )
λN

1

N
−1
Y
k=1

duN −1,k ,(5.1)

To match the historical notation for Whittaker functions, here and in the discussion of the spin
Whittaker functions we place the “variables” into the subscript of a Whittaker function, and the “index”
in the parentheses.

136where→N −1
QN(uN , uN −1 ) = eiλ(λPN

i=1 uN,i −

PN −1
i=1

uN −1,i )

N
−1
Y
i=1


exp −euN −1,i −uN,i − euN,i+1 −uN −1,i(5.2)
is known as the Baxter Q-operator. The function λ 7→ ψλ (uN ) is an entire functionof λ ∈ CN for all uN ∈ RN . For N = 1, we have ψλ (u) = eiλu . For N = 2, theWhittaker Rfunctions can be expressed through the (single-variable) Bessel K function
z x
1 ∞
xv
−x
Kv (z) = 2 −∞ e exp − 2 (e + e ) dx.
→N −1
For the Whittaker functions, QN(uN , uN −1 ) plays the role of a branching function
λN
like the single-variable sqW function Fν/µ (x) (2.15) (here x plays the same role as λN ,and ν, µ correspond to uN , uN −1 ). Note that the Whittaker functions are not indexed
by ordered sequences of numbers uN . Rather, in the Baxter Q-operator, the interlacing
condition among arrays uN −1 , uN is replaced by the “mild interlacing”. Namely, QN →N −1(5.2) decays doubly exponentially whenever uN,i+1 > uN −1,i or uN −1,i > uN,i .
The Whittaker functions satisfy the following analogue of the Cauchy identity due toBump and Stade [Bum89], [Sta02], [GLO08]:
Z
N
N
Y
Y
−u
−e N,NψλN (uN )ψν N (uN )duN,j =
Γ(iνj − iλk ).
(5.3)
e
RN

j=1

j,k=1

See also [Cor+14, (1.2)], [BC14, Section 4.2.1] for a generalization when one of the Whittaker functions is replaced by a certain integral coming from the limit of the torus productrepresentation of Macdonald polynomials:
Z
T Y
N
Y
Y
1
1
Γ(Yi − iνk ) ·ψν (uN )dν, (5.4)
θY (uN ) :=N
(2π) N ! 1≤A6=B≤N Γ(iνA − iνB )RN
i=1 k=1
where Y = (Y1 , . . . , YT ) ∈ RT . We refer to θY (uN ) as the dual Whittaker function.
Similar integral representations for dual spin Hall–Littlewood functions are found in[Bor17, Proposition 7.3], [BP18a, Section 7.3].
The Whittaker functions are eigenfunctions of the glN quantum Toda Hamiltonian
H2Toda , see formula (1.36) in the Introduction.
Convention on multiplicative notation. The papers [Cor+14], [OSZ14] use multiplicative parameters UN,i = euN,i ∈ R>0 instead of the additive ones. In multiplicative
notation, the integration in (5.1) and (5.3) is over the product measures of the form
Q dUm,i. It is convenient for us to adopt multiplicative notation throughout most of theUm,idiscussion of the spin Whittaker functions. We will often denote multiplicative variables
and parameters by capital letters.

5.1.2

Signatures in continuous space

In contrast with the usual Whittaker functions indexed by unordered N -tuples of reals,
the spin Whittaker functions will be indexed by nondecreasing sequences of real numbers.
137

Introduce the Weyl chamber of RN≥1 by
WN := {LN = (LN,i )1≤i≤N ∈ RN
≥1 : LN,N ≤ LN,N −1 ≤ . . . ≤ LN,1 }.

(5.5)

By W̊N denote the interior of the Weyl chamber with strict inequalities in (5.5).
Given two sequences LN −1 ∈ WN −1 and LN ∈ WN , we say that they interlace if
LN,i+1 ≤ LN −1,i ≤ LN,i ,for 1 ≤ i ≤ N − 1.

(5.6)

As in discrete setting, we denote interlacing by LN −1 ≺ LN . The interlacing relation isnaturally extended to sequences of the same length by dropping the last inequality in(2.1).
Q
dLN,kdL
We endow the Weyl chamber WN with the measure L N = N
k=1 LN,k . In most casesNdL

we do not explicitly indicate the integration domain WN when the measure L N is used.
N
Define the continuous Gelfand-Tsetlin cone as
N (N +1)/2

GTN := {LN = (Lk,i )1≤i≤k≤N ∈ R≥1: Lk+1,i+1 ≤ Lk,i ≤ Lk+1,i },(5.7)

which is the set of interlacing sequences L1 ≺ · · · ≺ LN . The set GTN is endowed withQ
dL
dL
the measure L N = 1≤i≤j≤N Lj,ij,i .
N

5.1.3

Spin Whittaker functions

We begin with a branching function from which we can recursively build spin Whittaker
functions. The branching function is an analogue of the skew polynomial evaluated at asingle variable.
Fix a deformation parameter S > 0 throughout the section. Let us denote
AS,X (u, v, z) :=1v S−X−1 u S+X−1 u 1−2S
1−
1−
1−,
B(S + X, S − X)
z
v
z

(5.8)

where 1 ≤ u < v < z are real, and |X| < S. Here B(·, ·) is the beta function (A.21).
Definition 5.1.1. Let |X| < S and k ≥ 1. The spin Whittaker branching functions are
given byfX (Lk ; Lk+1 ) := 1Lk ≺Lk+1



Lk+1,k+1 · · · Lk+1,1
Lk,k · · · Lk,1

−X Y
k
i=1

AS,X (Lk+1,i+1 , Lk,i , Lk+1,i ).

We now introduce the main object of the present section.
Definition 5.1.2 (Spin Whittaker functions). For N ≥ 1, consider parameters X1 , . . . , XN
and S such that |Xi | < S for all i. The spin Whittaker functions fX1 ,...,XN (LN ), LN ∈ WN ,are defined recursively by1
fX1 (L1,1 ) := L−X
(5.9)
1,1
138

for N = 1, and via the branching ruleZdL
fX1 ,...,XN −1 (LN −1 ) fXN (LN −1 ; LN ) N −1
fX1 ,...,XN (LN ) :=LN −1
LN −1 ≺LN

(5.10)

for N ≥ 2.
Example 5.1.3 (Two-variable spin Whittaker function). Let us compute the integral(5.10) for N = 2. Denote X 2 = (X, Y ), L2 = (u, u + α), where u ≥ 1, α > 0. Then
fX,Y (u, u + α)
1−2S Z u+αS−Y −1 (u(u + α))−Yuvu S+Y −1
Y −X−1
=
1−
1−
1−
v
dv
B(S + Y, S − Y )
u+α
u+α
v
u
Z 1
u−Y (u + α)S=(u + tα)−X−S (1 − t)S−Y −1 tS+Y −1 dt,
B(S + Y, S − Y ) 0

where we changed the variable as v = u + αt, t ∈ [0, 1]. The integral can now be evaluatedusing Euler’s representation of the Gauss hypergeometric function 2 F1 (A.24). Let us alsorename back z = u + α. We havez
S + X, S + Y
S −X−Y.
(5.11)
fX,Y (u, z) = (z/u) u
1−
2 F1
2S
–
u

When |1 − z/u| ≥ 1, the hypergeometric function in (5.11) should be understood in thesense of analytic continuation.
We remark that most of the properties of the spin Whittaker functions given below
in this section can be directly derived for N = 2 from known properties of the Gausshypergeometric function 2 F1 .
Proposition 5.1.4. For X N = (X1 , . . . , XN ) with |Xi | < S, the spin Whittaker function
fX N (LN ) is well-defined and continuous in LN ∈ WN .
In particular, we can first define fX N (LN ) for LN ∈ W̊N , and then extend to the whole
Weyl chamber by continuity. (Note that AS,X (u, v, z) (5.8) might have a singularity at
u = z.) The proof of Proposition 5.1.4 is based on the next two lemmas.
Lemma 5.1.5. Let `1 > 0 and let f (·) be a left continuous function on R≥1 . Then, we
have
Z `1
d`2
lim−
AS,X (`3 , `2 , `1 )f (`2 ) = f (`1 ).
(5.12)`2`3 →`1 `3
Proof. To compute the limit set `3 = `1 − δ for a small positive δ. After a change ofvariable `2 = `1 − δ(1 − `02 ), the integral in (5.12) becomes

S+X
Z 1
`1
1
0
d`2
(1 − `02 )S−X−1 `0S+X−1
f (`1 − δ(1 − `02 )).
2B(S + X, S − X) 0
`1 − δ(1 − `02 )Using the left continuity of f , we see that the integrand converges to (1−`02 )S−X−1 `0S+X−1
f (`1 )
2as δ → 0. The limiting integrand integrates to B(S + X, S − X), and so by the DominatedConvergence Theorem the lemma follows.
139Lemma 5.1.6. Let f : WN −1 → C be left continuous in each of LN −1,i . Define F : W̊N →
C as
Z
dL
(5.13)F (LN ) = f (LN −1 ) fX (LN −1 ; LN ) N −1 .
LN −1Then F is continuous and can be extended by continuity to WN .
Proof. For LN ∈ W̊N , the singularities of the integrand in (5.13) come only from thebranching function fX (LN −1 , LN ) and they are of the form
S−X−1S+X−1
LN,i+1
LN −1,i
,
or
1−
1−
LN,i
LN −1,ifor some i. Because |X| < S these singularities are summable. Therefore F , is continuous
inside the interior W̊N of the Weyl chamber.
To prove that F can be extended by continuity to WN we first define, from small
positive increments δ1 , . . . , δN −1 , the quantities di = δi +· · ·+δN −1 for each i = 1, . . . , N −1.
We aim to compute the limit
limδ1 ,...,δN −1 →0F (LN,N , LN,N −1 + dN −1 , . . . , LN,1 + d1 ),when some of the LN,i ’s are equal to each other. Before the limit, this function is equalto
Z LN,N −1 +δN −1
dLN −1,N −1
AS,X (LN,N , LN −1,N −1 , LN,N −1 + δN −1 )LN −1,N −1
LN,N
!X
QN −1
Z LN,1 +d2 +δ1L
dLN −1,1
N −1,i.
f (LN −1 ) QN i=1···
AS,X (LN,2 + d2 , LN −1,1 , LN,1 + d2 + δ1 )
L
(L
+
d
)
N −1,1
LN,2 +d2N,iii=1

For any i such that LN,i = LN,i+1 , make the change of variables LN −1,i = LN,i +di+1 −δi (1−`N −1,i ). As in the proof of Lemma 5.1.5, this removes all the corresponding singularities.
Therefore, the limit as δ1 , . . . , δN −1 → 0 exists, is finite, and can be computed using(5.12).
Proof of Proposition 5.1.4. For N = 1 the spin Whittaker function (5.9) is clearly continuous. Therefore, by Lemma 5.1.6, fX1 ,X2 (L2 ) is well defined and continuous on W2 .
Proceeding by induction on N , we get the result of Proposition 5.1.4.
The next corollary gives a Givental type representation of the spin Whittaker functions, obtained by writing down explicitly the recursive definition (5.10).
Corollary 5.1.7. We haveZ Y Qk−1 XkY
dLN −1i=1 Lk−1,ifX1 ,...,XN (LN ) =
.
A
(L
,
L
,
L
)
Qk
S,X
k+1,i+1
k,i
k+1,i
k+1
Xk
L
L
k,i
N
−1
i=1
1≤k≤N
1≤i≤k≤N −1(5.14)

Proof. Because the sequence of integrations as in (5.10) leading to fX N (LN ) is (absolutely) convergent, so is the integration over the Gelfand-Tsetlin array GTN −1 . The two
integration procedures give the same result by the Fubini–Tonelli theorem.
140

5.1.4

Dual Spin Whittaker functionse k ≺ Lk
In this section we define a dual family of functions. Given interlacing sequences Lof the same length k, introduce the dual spin Whittaker branching functions
1
ek ; Lk ) := 1 e
gY (LLk ≺LkΓ(S − Y )ek,k · · · L
ek,1L
Lk,k · · · Lk,1

!Y

×

ek,1
L
1−
Lk,1

k
Y
i=2

!S−Y −1(5.15)

ek,i , Lk,i , Lek,i−1 ).
AS,−Y (LFor pairs of interlacing sequences Lk−1 ≺ Lk , k ≥ 1, of different lengths, set
gY (Lk−1 ; Lk ) := gY ((1, Lk−1 ); Lk ).
Remark 5.1.8. One can also write gY as
−Y
Lk,1
e
gY (Lk ; Lk ) =Γ(S − Y )

ek,1
L
1−
Lk,1

where Lk = (`k−1 , Lk,1 ).

!S−Y −1ek ),
f−Y (`k−1 ; L(5.16)

Definition 5.1.9. Let N ≤ M and consider parameters Y1 , . . . , YM such that |Yi | < S
for all i. The dual spin Whittaker functions are defined recursively by
Z
eN

dL

e
e

g
(
L
)g
(
L
;
L
)
if N < M,
Y,...,Y
Y
N
N
N
1
M −1
M

eN
L
gY1 ,...,YM (LN ) = Z(5.17)
eN −1

dL


e
e
if N = M.
 gY1 ,...,YN −1 (LN −1 )gYN (LN −1 ; LN ) e
LN −1
In particular, for M = N = 1 we havegY (L) = gY (1; L) =L−Y (1 − L−1 )S−Y −1.
Γ(S − Y )The next two propositions explain that gY1 ,...,YM are well-defined as elements of the“dual” space of compactly supported continuous functions on the Weyl chamber WN .
Proposition 5.1.10. Let f (LN ) be a compactly supported continuous function on WN .
Then the functionZeeN ; LN )f (LN ) dLN ,LN 7→ gY (L(5.18)
LN
is also compactly supported and continuous.

141

Proof. We evaluate the integral (5.18) using expression (5.16) for gY as
Z

dLN,1
1
1+Y
LN,1 Γ(S − Y )

eN,1
L
1−
LN,1

!S−Y −1 Z

f (`N −1 , LN,1 )d`N −1ek ).
f−Y (`N −1 ; L`N −1

By Lemma 5.1.6, the integral in the variables `N −1 defines a family of continuous boundedeN , depending on LN,1 . The (improper) integral in LN,1 is convergent both
functions in L
eN,1 and ∞ (the latter because f vanishes for LN,1 large enough). This proves theat Lclaim.

Proposition 5.1.11. Let f (LN ) be a compactly supported continuous function. Then the
integralZdL
gY1 ,...,YM (LN )f (LN ) N
LN
is absolutely convergent.

Proof. This follows from Proposition 5.1.10 applied recursively after expanding gY1 ,...,YM
using the branching rules (5.17).

5.1.5

Convergence of the sqW functions as q → 1

Here and in the following subsection we establish that the spin Whittaker functions fX (LN )and gY (LN ) are scaling limits, as q → 1, of the spin q-Whittaker functions Fλ (x1 , . . . , xN )and F∗µ (y1 , . . . , yk ), respectively. Recall that they also depend on two parameters, q ∈ (0, 1)
and s ∈ (−1, 0).
First, this subsection we deal with the non-dual functions. Let us fix a scaling of allparameters.
Definition 5.1.12 (Scaling). We consider the following renormalization of parameters:xi = q Xi ,s = −q S ,λij = logq (1/Li,j ) .
(5.19)

We will assume throughout thatS > 0,|Xi | < S,and1 ≤ Li+1,j+1 ≤ Li,j ≤ Li+1,j

for all i, j. Therefore, the pre-limit quantities in (5.19) satisfy s ∈ (0, 1), xi ∈ (−s, −s−1 ),i+1i
and 0 ≤ λi+1
j+1 ≤ λj ≤ λj .
For any triple of real numbers 1 ≤ `3 ≤ `2 ≤ `1 , set ni := logq (1/`i ) (so 0 ≤ n3 ≤n2 ≤ n1 ).
Lemma 5.1.13. With the above notation, for any function f : Z → R we haven1X

n2 =n3f (n2 ) =

Z `1
`3


 d`2
1
f ( logq (1/`2 ) ),∆q (`3 , `2 , `1 )`2
142

(5.20)

where− log qZ min(`1 ,q−n2 −1 ) 0

log(q n1 `1 )d`2=∆q (`3 , `2 , `1 ) :=
`02
− log(q n3 +1 `3 )max(`3 ,q −n2 )log(`1 /`3 )if n3 < n2 < n1 ;if n3 < n2 = n1 ;if n3 = n2 < n1 ;if n3 = n2 = n1 .

(5.21)

When `3 = `1 , the integral in (5.20) is understood in the limiting sense.
Proof. This follows
 by observing that ∆q is the measure of intervals where the function`2 7→ logq (1/`2 ) is constant, and simultaneously `2 lies in the interval [`3 , `1 ].
The rescaled spin q-Whittaker functions are defined recursively as(q)fXN (LN −1 ; LN ) =

N
−1
Y

1
F N N −1 (xN );∆q (LN,k+1 , LN −1,k , LN,k ) λ /λscaling (5.19)
k=1
λ1

(q)

fX1 (L1,1 ) = x1 1(q)

fX1 ,...,XN (LN ) =Z

scaling (5.19)= q X1 blogq (1/L1,1 )c ;

(q)

(q)

fX1 ,...,XN −1 (LN −1 ) fXN (LN −1 ; LN )dLN −1,
LN −1

The next theorem is the main result of this subsection:
Theorem 5.1.14. We have(q)

lim fX1 ,...,XN = fX1 ,...,XN ,q→1(5.22)

uniformly on any compact subset of WN .
Pointwise convergence in (5.22) is a consequence of a simpler result stated in Lemma
2.2 of [BC16a] (reproduced as Lemma 5.1.25 in Section 5.1.9):(`q A ; q)∞= (1 − `)B−A ,q→1 (`q B ; q)∞lim(5.23)

for any ` ∈ (0, 1) and A, B > 0.
By (5.23) and through a repeated use of the identity
b+n
; q)∞
(q a ; q)nΓq (b)b−a (q
=
1
+
1
(1
−
q)
,
n=0
n≥1
(q b ; q)nΓq (a)(q a+n ; q)∞(5.24)

where Γq is the q-Gamma function (A.15), one readily gets the pointwise convergence(q)of the branching function fX (LN −1 ; LN ) to fX (LN −1 ; LN ). Nevertheless, for the finer
uniform convergence result of Theorem 5.1.14, a slightly more accurate analysis of ratiosof q-Pochhammer symbols appearing in the sqW functions is required. We postpone thistechnical discussion to Section 5.1.9. Let us summarize the main technical result provenin Section 5.1.9:
143

Proposition 5.1.15. Let f (LN −1 ) be a continuous function on WN −1 . Then for any
LN ∈ WN we haveZ
Z
dLN −1dL(q)
lim f (LN −1 ) fX (LN −1 ; LN )= f (LN −1 ) fX (LN −1 ; LN ) N −1 ,(5.25)
q→1
LN −1
LN −1
and the convergence is uniform on compact subsets of WN .
The continuous function f in Proposition 5.1.15 can also be replaced by a uniformly
converging sequence:
Corollary 5.1.16. Let f (q) (LN −1 ) be a sequence uniformly convergent as q → 1 on compact subsets of WN −1 to a continuous function f (LN −1 ). ThenZ
Z
dLN −1dL(q)(q)
lim f (LN −1 ) fX (LN −1 ; LN )= f (LN −1 ) fX (LN −1 ; LN ) N −1
q→1
LN −1
LN −1
and the convergence is uniform on compact subsets of WN .
Proof. This follows from Proposition 5.1.15 and the fact that for fixed LN ∈ WN , the(q)functions LN −1 7→ fX (LN −1 ; LN ) and LN −1 7→ fX (LN −1 ; LN ) are compactly supported on
WN −1 .
Proof of Theorem 5.1.14. For N = 1 we have
fX1 (L) = q X1 blogq (1/L)c −−→ L−X1 = fX1 (L),
(q)

q→1

uniformly with respect to L ≥ 1 varying in any compact domain. Corollary 5.1.16 thenimplies Theorem 5.1.14 by induction on N .

5.1.6

Convergence of the dual sqW functions as q → 1

We now establish the convergence of functions F∗ to the dual spin Whittaker functions g.
The scaling of parameters we adopt is that of Definition 5.1.12. For consistency with theprevious sections, dual functions will depend on y variables for which the scaling is
y i = q Yi ,|Yi | < S.

(5.26)

ek ≺ Lk define the rescaled dual spin Whittaker branching
For two interlacing arrays L
functions
!
k
Y
1
(q) e
S−Yk
gYk (LF∗λk /λek (yk ), (5.27)
k ; Lk ) = (1 − q)
e
e
∆
(
L
,
L
,
L
)
scaling(5.19),(5.26)
k,j−1
q
k,j
k,j
j=1
ek,0 = ∞, and ∆q is given by (5.21). In particular, the rescaled
where, by agreement, L
one-variable function is (assuming L > 1 and q close enough to 1)(q)(q)
gY (L) = gY (1; L) = (1 − q)S−Y(q S−Y ; q)blogq (1/L)c
1
q Y blogq (1/L)c .
(− log q) (q; q)blogq (1/L)c
144

(q)

(q)

For interlacing arrays of different lengths Lk−1 ≺ Lk , we set gY (Lk−1 ; Lk ) = gY ((1, Lk−1 ); Lk ),as before. Define the rescaled dual spin q-Whittaker functions recursively as
Z
e

(q)

eN ; LN ) dLNeN )g(q) (L
if N < M,
g
(
L

YM
Y1 ,...,YM −1

eN
L
(q)
gY1 ,...,YM (LN ) = Z
e


(q)

eN −1 )g(q) (LeN −1 ; LN ) dLN −1g(Lif N = M.
Y1 ,...,YN −1
YN

eN −1
L
The next result establishes a weak convergence of rescaled branching functions g(q) .

Theorem 5.1.17. Let f (LN ) be a compactly supported continuous function on WN . Then
Z
Z
dLN
(q) e
eN ; LN )f (LN ) dLN ,lim gY (LN ; LN )f (LN )= gY (L(5.28)
q→1
LN
LN

eN .
and the convergence is uniform with respect to L(q) e
Proof. We start by rewriting the branching function gY (LN ; LN ) as (this follows fromstraightforward algebraic manipulations with (5.27))

q

Y λk1

(q S−Y ; q)λk −λek
11(q; q)λk −λek
11

(1 − q)S−Y
(q)
ek ).
f−Y (`k−1 ; L
ek,1 , Lk,1 , ∞)∆q (LThe integral in the left-hand side of (5.28) becomes!Z
Z
S−Y(q S−Y ; q)λk −λekd`k−1 (q)
dLk,1
(1
−
q)
k
1
1
e )f (`k−1 , Lk,1 ).
f (` ; Lq Y λ11
ek,1 , Lk,1 , ∞)Lk,1(q; q)λk −λek ∆q (L`k−1 −Y k−1 k1
1(5.29)(q)e
The inner integral involving the function f−Y is uniformly (with respect to Lk ) convergent
to
Z
d`k−1
ek )f (`k−1 , Lk,1 )f−Y (`k−1 ; L`k−1
by virtue of Proposition 5.1.15. On the other hand, the term inside the parentheses in(5.29) is uniformly convergent toS−Y −1L−Yk,1e
1 − Lk,1 /Lk,1,
Γ(S − Y )ek,1 . Moreover, the term inside the parentheses is absowhen Lk,1 is kept away from L
ek,1 /Lk,1 )S−Y −1 when Lk,1 approaches L
ek,1 , thanks tolutely bounded by const × (1 − L
Lemma 5.1.26. Since the resulting term after the q → 1 limit coincides with the expreseN ; LN ), we are done.
sion (5.16) for the dual branching function gY (LSimilarly to Corollary 5.1.16, we can let the test function f depend on q:
145

Corollary 5.1.18. Let f (q) (LN ) converge, as q → 1, to a compactly supported continuousfunction f (LN ), uniformly on WN . Then
Z
Z
dLN
(q) e
(q)
eN ; LN )f (LN ) dLN= gY (Llim gY (LN ; LN )f (LN )
q→1
LN
LN
eN .
and the convergence is uniform with respect to L

5.1.7

Properties of the spin Whittaker functions

In this subsection we describe the properties of the spin Whittaker functions which follow
in the q → 1 limit from the corresponding properties of the spin q-Whittaker functions.
Proposition 5.1.19 (Symmetry and shifting). The spin Whittaker function fX1 ,...,XN (LN )is symmetric in the Xi ’s for all LN ∈ WN . They also satisfy the shifting property:
fX1 ,...,XN (aLN ) = a−X1 −...−XN fX1 ,...,XN (LN ),a > 1.

Proof. The symmetry follows from the corresponding symmetry of the sqW polynomial Fλ (x1 , . . . , xN ), which ultimately is a consequence of the Yang–Baxter equation.
The shifting property can either be deduced from Proposition 2.3.7, or obtained in asimilar way by noting that the branching spin Whittaker functions themselves satisfy
fX (aLk ; aLk ) = a−X fX (Lk ; Lk ).
We now turn to Cauchy type identities for the spin Whittaker functions.
Theorem 5.1.20 (Skew Cauchy type identity). Assume |X|, |Y | < S and X + Y > 0.
eN we haveThen, for any LN −1 , L
Z

eN ; LN ) dLNfX (LN −1 ; LN )gY (LLN

Γ(X + Y )Γ(2S)=Γ(S + X)Γ(S + Y )ZeeN −1 ; LeN )gY (LeN −1 ; LN −1 ) dLN −1 (5.30)
fX (L
eN −1
L

and, when N = 1 we haveZ
e1,1 ; L1,1 ) dL1,1 = Γ(X + Y ) fX (Le1,1 ).
fX (L1,1 )gY (LL1,1
Γ(S + X)

(5.31)

Proof. We first observe that (5.31) is equivalent to the integral representation of B(S −Y, X + Y ).
In order to prove the general case (5.30) we use Corollaries 5.1.16 and 5.1.18. Take acompactly supported continuous test function φ(LN −1 ), and set
ZZ
dLN −1eeN −1 ; LN −1 )φ(LN −1 ) dLN −1 .
Φf (LN ) := φ(LN −1 )fX (LN −1 ; LN ), Φg (LN −1 ) := gY (L
LN −1
LN −1
146

(q)

(q)

(q)

Analogously define Φf and Φg by substituting respectively fX and gY with fX and(q)
gY in the above formulas. It follows from the skew Cauchy Identity for sqW functions(theorem 2.3.15) that
Z

Z

eN −1
dL
(q) e
(q) e
e
(
L
;
L
)Φ
)
.
fX (L
N −1
N
N −1
g
eN −1
L
(5.32)(q)
By Corollary 5.1.18 we have Φg → Φg uniformly, and further Φg is compactly supported
and continuous by Proposition 5.1.10. This implies, by Corollary 5.1.16, that the righthand side of (5.32) converges to
dLN
(q)
(q) e
Φf (LN )gY (LN ; LN )LN

Γq (X + Y )Γq (2S)=
Γq (S + X)Γq (S + Y )Γ(X + Y )Γ(2S)Γ(S + X)Γ(S + Y )ZeeN )Φg (LeN −1 ) dLN −1 .
eN −1 ; L
fX (L
eN −1
L

The integral in the left-hand side of (5.32) is absolutely convergent when X +Y > 0. Since(q)Φf → Φf uniformly by Proposition 5.1.15, Corollary 5.1.18 implies that the left-hand sideof (5.32) converges toZ
eN ; LN ) dLN .
Φf (LN )gY (LLN
Since the function φ was arbitrary, equality (5.30) follows.

Corollary 5.1.21 (Full Cauchy type identity). Let N ≤ M and |Xi |, |Yj | < S, Xi +Yj > 0
for all i, j. We have
!
Z
M
N
Y
Y
dL
Γ(X1 + Yj )Γ(Xi + Yj )Γ(2S).
fX1 ,...,XN (LN ) gY1 ,...,YM (LN ) N =LN
Γ(S + X1 ) i=2 Γ(S + Xi )Γ(S + Yj )j=1

(5.33)

Proof. Immediately follows from Theorem 5.1.20 and the branching rules for the functions
f, g.
We also have an identity involving a single spin Whittaker function:
Proposition 5.1.22. Let |Xi | < S. Then we haveZ

LN,N =1

fX1 ,...,XN (LN )

N
−1 
Y
j=1

LN,j+1
1−
LN,j

2S−1

dLN,j
Γ(S + X1 ) · · · Γ(S + XN )=.
1+S
Γ(SN + X1 + · · · + XN )LN,jProof. This is a scaling limit of Proposition 2.3.9.
We now consider the scaling limits of eigenrelations for the sqW functions stated in
Theorems 2.6.11 and 2.6.12. This produces two operators acting in the Xi variables
which are diagonal in the spin Whittaker functions. For the next definition we use theshift operator
TX f (X) := f (X + 1).
(5.34)
147

Definition 5.1.23. For any N ≥ 1 set
D1 :=

N Y
N
X
Xi + Si=1 j=1
j6=iXi − X j

D1 :=

TXi ,

N Y
N
X
Xi − Si=1 j=1
j6=iXi − X j

TX−1.
i

The next proposition represents a partial generalization of eigenrelations satisfied by
Whittaker functions (e.g., see [KL01]).
Proposition 5.1.24 (Eigenrelations for spin Whittaker functions). We have
D1 fX1 ,...,XN (LN ) = L−1
N,N fX1 ,...,XN (LN ),D1 fX1 ,...,XN (LN ) = LN,1 fX1 ,...,XN (LN ).
Proof. We easily see that operators D1 , D1 are limiting forms of D1 , D1 (Definition 2.6.9)−λ1
under the scaling (5.19). At the same time we have q λN → L−1→ LN,1 underN,N and q
the same scaling. Therefore, (2.87), (2.91) and convergence (5.22) imply the claimed
eigenrelation.

5.1.8

Formal reduction to the usual Whittaker functions

Just like the sqW polynomials reduce to the q-Whittaker polynomials setting s = 0, it
should be possible to prove that, under the correct scaling, our spin Whittaker functions
converge to the Whittaker functions. An evidence for this is suggested by the following
computation.
Set
Lk,i = S k+1−2i euk,i ,Xk = −iλk ,(5.35)
then, in the limit S → ∞ we have
Xk+1
Lk,k · · · Lk,1
Lk+1,k+1 · · · Lk+1,1
S−Xk+1 −1

Lk,i
1−
Lk+1,iS+Xk+1 −1
Lk+1,i+1
1−
Lk,i

1−2S
Lk+1,i+11−
Lk+1,i

1

4S S 2 B(S + X, S − X)(−−−−−−→ exp iλk

k+1
X
i=1

uk+1,i −−−−−−−→ exp −euk,i −uk+1,i ;−−−−−−→ exp −euk+1,i+1 −uk,i ;

k
X
i=1

uk,i

!)

;

(5.36)−−−−−−→ 1;√−−−−−−→ 2 π.

All the limits in (5.36) are straightforward (note that the last one requires the Stirling
approximation). Thus, the branching function f, rescaled by a factor depending solely148

on S, converges locally uniformly to the Baxter Q-operator QN →N −1 (5.2) for the usual
Whittaker functions: N2−1
4π
scaling (5.35)→N −1
fXN (LN −1 ; LN ) −−−−−−−→ QN(uN , uN −1 ).
λN
S
S→∞
S16
These computations suggest that the same type of convergence should hold for the fullfunctions. Namely, under (5.35) and as S → +∞, the spin Whittaker functions fX N (LN )N (N −1)

rescaled by (4S −1 π/16S ) 4 should converge to the usual Whittaker functions ψλN (uN ).
A proof of this convergence would require a finer analysis to justify the exchange of the
S → +∞ limit and integration, and goes beyond the scope of this thesis.

5.1.9

Proof of Proposition 5.1.15

Lemma 5.1.25 ([BC16a], Lemma 2.2). Let A, B > 0. Then(`q A ; q)∞= (1 − `)B−A ,q→1 (`q B ; q)∞lim(5.37)

uniformly in ` belonging to any compact subset of (0, 1).
Note that the uniformity in ` in (5.37) is not claimed in [BC16a] but easily follows fromthe uniformity of all Taylor expansions involved in the proof in the cited paper (which we
do not reproduce).
Lemma 5.1.26. Let A, B > 0. Then, for all n ∈ Z≥1 and all q ∈ ( 21 , 1), we have(q A+n ; q)∞≤ c (1 − q n )B−A ,(q B+n ; q)∞(5.38)

where c is a constant independent of q, n.
Proof. Set q = e−ε . The result of the Lemma is restated, taking the logarithm of bothsides of (5.38), asX(1 − e−ε(A+n+k) )− (B − A) log(1 − e−εn ) ≤ c0 ,log
−ε(B+n+k)(1 − e)k≥0(5.39)

for all ε ∈ (0, − log 2) and a constant c0 independent of ε, n. Using Lagrange mean value
theorem, we can rewrite the generic term of the infinite sum aslog(1 − e−ε(A+n+k) )ε,= (A − B) ε(et +n+k)−ε(B+n+k)(1 − e)e k−1where numbers e
tk belong to the interval (min(A, B), max(A, B)). We show that for any
positive bounded sequence {tk }k ⊂ (0, M ), with M fixed, the quantityXε+ log(1 − e−εn )(5.40)
ε(t
+n+k)
k
e
−
1
k≥0
149

is absolutely bounded uniformly in ε and n and this would prove (5.39) and hence (5.38).
To evaluate the infinite sum over k we fix a positive constant δ and distinguish two cases.
Case 1, k ≥ δ/ε. We use the estimate
ε
ε
−εk
≤
e
,
eε(tk +n+k) − 1
1 − e−δthat implies, summing over k,
X
k≥δ/ε

ε

eε(tk +n+k) − 1

≤

ε

1

eδ − 1 1 − e−ε≤2 log 2.
eδ − 1(5.41)

Case 2, k < δ/ε. In this case we use again Lagrange mean value theorem to express thedenominator of the generic term of the summation of (5.40) asεeε(tk +n+k) − 1=e−εξk,n,tk + n + k

for some ξk,n ∈ (0, tk + n + k).

This implies the boundse−ε(M +n+k)ε
1≤ ε(t +n+k)≤.
k
M +n+ke−1n+k(5.42)

We focus first on the lower bound given by the first inequality in (5.42). Summing overk we findZ δ/ε −ε(M +n+k)Z δ+ε(M +n) −k0
Z δ+ε(M +n) δ/ε −ε(M +n+k)
X
e
e
1
e
0
≥
dk =
dk ≥
− 1 dk 0 ,
0
0
M
+
n
+
k
M
+
n
+
k
k
k
0
ε(M +n)ε(M +n)k=0which gives
δ/ε
X
k=0

ε
eε(tk +n+k) − 1

≥ log 1 +δε(M + n)− δ.

(5.43)

We turn now our attention to the second inequality in (5.42) and, since
δ/ε
X

1
≤
n+k
k=0

we obtainδ/εX

Z δ/ε+n
n



ε

dk
,
k − 1/2

δ≤ log 1 +
ε(t
+n+k)
e k
−1
ε(n − 1/2)
k=0



.

(5.44)

Combining results obtained from the analysis of cases k ≥ δ/ε and k < δ/ε in (5.41),(5.43) (5.44) we can finally write




δ
δ
−εn
−εn
log (1 +)(1 − e ) +O(δ) ≤ (5.40) ≤ log (1 +)(1 − e ) +O(δ).
ε(M + n)ε(n − 1/2)
This concludes our proof since the arguments of the logarithms in the left and right-handside are bounded functions for ε ∈ (0, log 2) and n ≥ 1.
150

For the next lemma we define the quantity1(q S−X ; q)n1 −n2 (q S+X ; q)n2 −n3 (q; q)n1 −n3,∆q (`3 , `2 , `1 ) (q; q)n1 −n2(q; q)n2 −n3 (q 2S ; q)n1 −n3
where we assumed 1 ≤ `3 ≤ `2 ≤ `1 and ni = logq (1/`i ) . Here ∆q is defined in (5.21).
(q)
We think of AS,X as a q-deformation of AS,X (5.8).
(q)AS,X (`3 , `2 , `1 ) =Lemma 5.1.27. For any continuous function f (`2 ) we have
lim

q→1

Z `1
`3

Z `1

d`2
(q)
=
f (`2 )AS,X (`3 , `2 , `1 )`2

f (`2 )AS,X (`3 , `2 , `1 )

`3

d`2
,
`2

(5.45)

uniformly for any `3 ≤ `1 bounded away from ∞.
Proof. Fix small positive δ. We will prove our claim distinguishing two cases, based on
the distance between `3 and `1 .
Case 1, `1 − `3 > δ. The integral in the left-hand side of (5.45) can be decomposed as
Z `1
`3

=

Z `1 −δ/2`3 +δ/2+

Z `3 +δ/2

+

`3

Z `1

.

`1 −δ/2

When `3 + δ/2 ≤ `2 ≤ `1 − δ/2 , by virtue of Lemma 5.1.25, we haveZ `1 −δ/2d`2(q)f (`2 )AS,X (`3 , `2 , `1 )−−→`2 q→1`3 +δ/2

Z `1 −δ/2

f (`2 )AS,X (`3 , `2 , `1 )`3 +δ/2d`2,`2

uniformly.
On the other hand, when `3 ≤ `2 < `3 + δ/2 we use estimates
if n3 < n2 ,CAS,X (`3 , `2 , `1 )

S+X
(q)
AS,X (`3 , `2 , `1 ) ≤
,
C
1−q

if n3 = n2 .
∆q (`3 , `2 , `1 ) 1 − `3 /`1

valid for some constant C independent of q of `2 and that can be deduced using Lemma 5.1.26
and identity (5.24). This implies thatZ `3 +δ/2
`3

d`2
(q)
= O(δ) + O
f (`2 )AS,X (`3 , `2 , `1 )
`2



S+X

.

S−X

.

1−q
δ

In an analogous fashion one can also show that
Z `1

d`2
(q)
f (`2 )AS,X (`3 , `2 , `1 )= O(δ) + O`2`1 −δ/2

This concludes the proof of (5.45) when `1 − `3 > δ.
151



1−q
δ

Case 2, `1 − `3 ≤ δ. Assuming δ is very small, for any `2 ∈ [`3 , `1 ], we can write, bycontinuity, f (`2 ) = f (`1 ) + o(1), where o(1) tends to 0 as δ → 0. Thus, we have
Z `1
n1
X
d`2
(q)
(f (`1 ) + o(1)) q −(S+X)(n1 −n2 ) ϕq,qS+X ,q2S (n1 − n2 |n1 − n3 )f (`2 )AS,X (`3 , `2 , `1 )
=
`
2
`3
n =n
2

3

= f (`1 ) + o(1),and by Lemma 5.12 this concludes the analysis of the case `1 − `3 ≤ δ.
Since all the estimates we provided are controlled as functions of δ, the convergencein (5.45) is uniform provided that `1 stays bounded.
Proof of Proposition 5.1.15. The integral in the left-hand side of (5.25) is equal toZ LN,N −1
dLN −1,N −1 (q)A (LN,N , LN −1,N −1 , LN,N −1 ) · · ·LN −1,N −1 S,X
LN,N

X
Z LN,1dLN −1,1 (q)LN −1,N −1 · · · LN −1,1···f (LN −1 )A (LN,2 , LN −1,1 , LN,1 )LN −1,1 S,X
LN,N · · · LN,1
LN,2
and we can take the q → 1 limit in each of the N − 1 integrals using Lemma 5.1.27. Thisestablishes the convergence to the right-hand side of (5.25) as q → 1, uniformly on any
compact subset of WN .

5.2

Spin Whittaker Processes and beta polymers

In this section define spin Whittaker processes, and establish their connection with twobeta polymer type models introduced in [BC16a] and [CMP19], respectively.

5.2.1

Spin Whittaker processes

The definition of spin Whittaker processes is a straightforward analogy of the discrete level
F/G processes (Definition 3.1.3). The key role is played by the Cauchy type identities(established for spin Whittaker functions in Section 5.1.7).
Definition 5.2.1. Set X = (X1 , . . . , XN ) and Y = (Y1 , . . . , YT ), with |Xi |, |Yj | < S and
Xi + Yj > 0 for all i, j. The (ascending) spin Whittaker process is the probability measure
on interlacing sequences LN (T ) = (Lk,i (T ))1≤i≤k≤N (that is, on the Gelfand-Tsetlin coneQ
dL
dL
GT N (5.7)) with the following density with respect to the measure L N = 1≤i≤j≤N Lj,ij,i :N

PX;Y (LN ) =fX1 (L1 )fX2 (L1 ; L2 ) · · · fXN (LN −1 ; LN )gY (LN ).
Π(X; Y)(5.46)The normalizing constant in (5.46) follows from the Cauchy identity of Corollary 5.1.21:!
TNY
Γ(X1 + Yj ) Y Γ(Xi + Yj )Γ(2S)
Π(X; Y) =
.
Γ(S
+
X
)
Γ(S
+
X
)Γ(S
+
Y
)
1
i
j
j=1
i=2
152

For the next result we denote the ascending sqW/sqW process, subject to the rescaling(5.19), (5.26), by
(q)
(q)
(q)
(q)
fX1 (L1 )fX2 (L1 ; L2 ) · · · fXN (LN −1 ; LN )gY (LN )(q),
PX;Y (LN ) =(q)Π (X; Y)

where the normalization constant is (cf. (5.32))
!
T
N
Y
Y
Γ
(X
+
Y
)
Γ
(X
+
Y
)Γ
(2S)
q
1
j
q
i
j
q
Π(q) (X; Y) =.
Γq (S + X1 ) i=2 Γq (S + Xi )Γq (S + Yj )j=1
Theorem 5.2.2. Under the scaling (5.19), (5.26), the ascending sqW/sqW process converges weakly to the spin Whittaker process(q)

PX;Y −−→ PX;Y .
q→1(5.47)

Proof. For any continuous bounded test function φ(LN ) on GT N we haveZ
Z dL
1dLN (q)(q)(q)
N −1 (q)(q)
E (φ) = (q)gY (LN )fX1 (L1 )fX2 (L1 ; L2 ) · · · fXN (LN −1 ; LN )φ(LN ).
Π (X; Y)
LN
LN −1
(5.48)The integral is absolutely convergent and as a consequence of Corollaries 5.1.16 and 5.1.18
it converges to the average E(φ) with respect to the spin Whittaker process.
Remark 5.2.3. Developing the argument sketched in Section 5.1.8 it should be possible
to show that the spin Whittaker process of Definition 5.2.1 converges to the α-Whittakerprocess from [Cor+14], [BC14]. In this case the correct way to rescale the random variables
Lk,i (T ) is
Lk,i (T ) = S T +k+1−2i euk,i (T ) .
(5.49)In the limit S → ∞ the process {uk,i (T ) : 1 ≤ i ≤ k ≤ N } should be then described by
the density
N →N −1
2→1(uN , uN −1 )θY (uN )Q1→0iX1 (u1 , 0)QiX2 (u2 , u1 ) · · · QiXN,
(5.50)
QN QT
i=1
j=1 Γ(Xi + Yj )where Qk+1→k and θY are given in (5.2) and (5.4), respectively.

5.2.2

Strict-weak beta polymer model

We will now recall the strict-weak beta polymer introduced in [BC16a].
Definition 5.2.4. Let Bi,j ∼ B(Xi + Yj , S − Yj ) be a family of independent beta randomvariables. The strict-weak beta polymer model partition function Z(i, j), i ≥ 1, j ≥ 0, is
the random function satisfying the recurrencefor 1 < i ≤ j;Z(i, j) = Z(i, j − 1)Bi,j + Z(i − 1, j − 1)(1 − Bi,j )Z(1, j) = Z(1, j − 1)B1,j
for j > 0;Z(i, 0) = 1
for i > 0.
153

Note that all the partition functions Z(i, j) belong to (0, 1]. In particular, the probability distribution of the strict-weak beta polymer is completely determined by the joint
moments.
ur(Section 3.3.5).
Proposition 5.2.5. Recall the q-Hahn vertex model height function Hq-Hahn
ur
Hq-Hahn (i,j)(q)(q)
Define Z (i, j) = q. Then, under the scaling (5.19), Z converges weakly tothe strict-weak beta polymer partition function:

Z (q) (i, j) −−→ Z(i, j).
q→1

Proof. This result is equivalent to Proposition 2.1 of [BC16a] in the homogeneous case
Xi = X, Yj = Y for all i, j. One can easily check that the proof given there also works in
our inhomogeneous setting.
Theorem 5.2.6. The marginal process {Lk,k (T )−1 : k = 1, . . . N, T ≥ N } of the spin
Whittaker process PX,Y is equivalent in distribution to the strict-weak beta polymer partition functions model {Z(k, T ) : k = 1, . . . , N, T ≥ N }.
Note that since Lk,k (T ) ∈ R≥1 , we have Lk,k (T )−1 ∈ (0, 1], which agrees with the rangeof the beta polymer partition functions.
Proof of Theorem 5.2.6. This is a direct consequence of Theorems 3.3.16 and 5.2.2 andProposition 5.2.5. Indeed, the last row marginal of the sqW/sqW process is matched tothe q-Hahn vertex model height function, and in the limit q → 1 this implies that thelast coordinate marginal of the spin Whittaker process is matched to the beta polymer
Z(i, j).
Let us make two remarks on this result.
Remark 5.2.7. A weaker version of Theorem 5.2.6 that matches Lki ,ki (T )−1 and Z(ki , T )for each single time T can alternatively be proved using moment formulas. Namely, theeigenoperators of Definition 5.1.23 may be used to extract multiple integral formulas forthe joint moments of Lki ,ki (T )−1 under the spin Whittaker processes. These formulas canthen be matched to the ones for the joint moments of the beta polymer. The latter inthe homogeneous case are obtained in [BC16a, Proposition 3.4], and their inhomogeneous
generalization is rather straightforward, cf. [Pet19, Proposition 6.1].
Remark 5.2.8. It was noticed in [BC16a, Remark 1.5] that under the scaling Z(i, T ) =S T −i+1 z(i, T ) the process z(i, T ) converges, when S → ∞, to the strict-weak gamma
polymer model introduced by Seppäläinen in an unpublished note and studied in [OO15],[CSS15]. This scaling of polymers corresponds to the scaling (5.49) for the full spin
Whittaker process. As S goes to infinity, Theorem 5.2.6 turns into the matching between
strict weak gamma polymer model and α-Whittaker process that was originally discoveredin [OO15]. This observation is another piece of evidence supporting the formal S → +∞scaling described in Section 5.1.8.

154

5.2.3

Another beta polymer type model

Let us now recall the beta polymer type model which was introduced in [CMP19]. We
employ notation from Chapter A.
e j), for i, j ∈ Z≥0 is defined by the recurrence
Definition 5.2.9. The random function Z(i,1for j = 0,




Z(1,
e j − 1)Be1,j
for i = 1,eZ(i, j) => e> ee j − 1) > Z(ie − 1, j),Z(i, j − 1) + (1 − Wi,j
)Z(i − 1, j)
if Z(i,
Wi,j




< e
< e
e j − 1) < Z(ie − 1, j),(1 − Wi,j)Z(i, j − 1) + Wi,jZ(i − 1, j)
if Z(i,(5.51)−1e
where B1,j ∼ B (X1 + Yj , S − Yj ) are independent inverse beta random variables, and
!
e
e
Z(i
−
1,
j)
−
Z(i
−
1,
j
−
1)
>
, X + Yj , S − Yj ,Wi,j
∼ NBB −1 2S − 1,
e j − 1) − Z(ie − 1, j − 1) iZ(i,!
(5.52)
e
e
Z(i,
j
−
1)
−
Z(i
−
1,
j
−
1)
<
, X + Yj , S − Xi .
Wi,j
∼ NBB −1 2S − 1,
e − 1, j) − Z(ie − 1, j − 1) iZ(ie j − 1) 6= Z(ie − 1, j) with probability one.
For i > 1, j > 0, we have Z(i,Proposition 5.2.10. Recall the 4 φ3 vertex model height function Hφul (i, j) (Section 3.3.6).
ulDefine Ze(q) (i, j) = q −Hφ (i,j) . Then, under the scaling (5.19), Z (q) converges weakly to theeprocess Z:
e j).
Ze(q) (i, j) −−→ Z(i,q→1Proof. In the homogeneous case Xi = 0, Yj = Y for all i, j, this was proven in [CMP19].
The same argument also essentially applies to the inhomogeneous case, and we will not
repeat the computations here. The non-trivial part of the proof is to understand how<>
the Xi parameters appear in the definition of Wi,j
, Wi,j. The interested reader can checkthe validity of our statements starting from (3.33) and reproducing the computations ofSection 4.3 of [CMP19].

Theorem 5.2.11. The marginal process {Lk,1 (T ) : k = 1 . . . N, T ≥ N } of the spin White T ) : k = 1, . . . N, T ≥taker process PX,Y is equivalent in distribution to the process {Z(k,N }.

Proof. This is established similarly to Theorem 5.2.6 by combining the matching of Theorem 3.3.20 with the q → 1 scaling limits.

5.2.4

Reduction to log-gamma polymer

e j) of Definition 5.2.9 reduces, as S → +∞, to the wellHere we show that the model Z(i,known log-gamma polymer model introduced in [Sep12]. This proof is more involved than
the rather straightforward observation for the strict-weak beta polymer (Remark 5.2.8).
155Definition 5.2.12 (Log-gamma polymer). Let {gi,j : i, j ∈ Z≥1 } be a sequence of independent inverse gamma random variables, gi,j ∼ Gamma−1 (Xi + Yj ) with density (A.22).
The random function ze defined by the recurrencez (i − 1, j) + ze(i, j − 1)) if i, j ≥ 1 and i + j ≥ 3;
gi,j (eze(i, j) = g1,1
(5.53)
if i = j = 1;


0
else,
is the point-to-point log-gamma polymer partition function.

The log-gamma polymer model was introduced (with a proof of its exact solvability) by Seppäläinen in [Sep12]. One can view ze(i, j) as a partition function of up-right
directed paths from (1, 1) to (i, j), where the weight of each path equals the productof the quantities gi0 ,j 0 along the path. In [Cor+14] the log-gamma polymer model was
given a powerful combinatorial interpretation using Kirillov’s geometric RSK (RobinsonSchensted-Knuth) algorithm. This showed the distributional matching of the log-gamma
polymer with a marginal of the Whittaker process (5.50).
The next statement shows that the log-gamma polymer model can be obtained in a
S → +∞ scaling limit from the beta polymer like model of Definition 5.2.9. Modulo
Remark 5.2.3, this together with Theorem 5.2.11 produces an alternative derivation of
the results of [Cor+14].
e j) = S j+i−1e
Proposition 5.2.13. Consider the scaling Z(i,z (S) (i, j) of the process from
Definition 5.2.9. Then the rescaled process ez (S) converges weakly to the log-gamma polymer:ez (S) (i, j) −−−→ ze(i, j).
S→∞

Proof. We argue by induction. When i = j = 1, then ez (S) (1, 1) = S −1 B −1 (X1 +Y1 , S −Y1 ).
In the large S limit this converges to Gamma−1 (X1 + Y1 ), which is precisely ze(1, 1).
Fix i, j and assume that for all i0 , j 0 such that i0 +j 0 < i+j the convergence ez (S) (i0 , j 0 ) →
>
<
0 0
−1
e
z (i , j ) holds. Let us compute the densities of random variables S Wi,j and S −1 Wi,j,>that are rescalings of (5.52), in the large S limit. We show the computations only for Wi,j>since the other case is very similar. The density of S −1 Wi,j(depending on the variablex ∈ (0, 1)) is, from (5.52) and (A.23), equal to
1 Xi +Yj +1x1
1 − SxΓ(Xi + Yj )

S−Yj −1

Γ(S + Xi )X
S i +Yj Γ(S − Yj )× (1 − pS )

2S−1

2 F1





1
2S − 1, S + Xi
pS 1 −, (5.54)
S − Yj
–
Sx

where
pS =

e − 1, j) − Z(ie − 1, j − 1)Z(iSez (S) (i − 1, j) − ez (S) (i − 1, j − 1)ze(i − 1, j)=∼ p :=(S)(S)
e j − 1) − Z(ie − 1, j − 1)Sez (i, j − 1) − ez (i − 1, j − 1)ze(i, j − 1)
Z(i,
156

is smaller than 1.
The limit of the first few factors in (5.54) is straightforward:
1 Xi +Yj +1 − x1
1 S−Yj −1
1 Xi +Yj +1
1 − SxeΓ(S + Xi )
x
x
−
−
−
−
−
→
.
Γ(Xi + Yj )S Xi +Yj Γ(S − Yj ) S→∞
Γ(Xi + Yj )To compute the limit of the Gaussian hypergeometric function, we use the Euler transformation




a, b
c − a, c − b
c−a−bz = (1 − z)
z ,
2 F1
2 F1
c, –
c
–

so that the remaining terms in (5.54) become2S−1 pS −Xi −Yj
1 − pS
1−S − Yj + 1, −Xi − Yj
1 − pS +.
pS 1 −2 F1pS
S − Yj ,–1 − pS + Sx
Sx
Sx
We have

1 − pSpS
1 − pS + Sx

2S−1 1 − pS +2p
pS −Xi −Yj−−−−−→ e− (1−p)x (1 − p)−Xi −Yj ,S→∞
Sx

whereas





1
−S − Yj + 1, −Xi − Yj
−Xi − Yj−−−−−→ 1 F0
pS 1 −− p = (1 + p)Xi +Yj .
2 F1
S − Yj ,––S→∞Sx
Our computations imply that>S −1 Wi,j−−−−−→
S→∞ez (i − 1, j) + ez (i, j − 1)Gamma−1 (Xi + Yj ).
ez (i, j − 1) − ez (i − 1, j)<
Essentially repeating the computations for S −1 Wi,j
, we obtain<
S −1 Wi,j−−−−−→
S→∞ez (i − 1, j) + ez (i, j − 1)Gamma−1 (Xi + Yj ).
ez (i − 1, j) − ez (i, j − 1)

Thus, we see that in the scaling limit as S → +∞, the beta polymer like model recurrence relation (5.51) becomes (5.53), the recurrence for the log-gamma polymer partition
functions. This completes the proof.

5.3

Deformed quantum Toda system

In this section we consider the scaling limit of the Pieri rule (2.36) which states that the
spin q-Whittaker polynomials Fλ (x1 , . . . , xN ) are eigenfunctions of an operator acting on
the label λ. This scaling limit leads to an eigenoperator for the spin Whittaker functions.
This operator acts as a second order differential operator in the (additive versions of the)variables LN . We call is the S-deformed quantum Toda Hamiltonian. Our scaling of thePieri rules are inspired by [GLO12] where the Pieri rule for the q-Whittaker polynomials
was understood as a discretization of the (undeformed) quantum Toda Hamiltonian.
157

5.3.1

Refined Pieri operators

P∗
We start by refining the Pieri operator (HsHL f )(µ) =λ f (λ) Fλ0 /µ0 (v) introduced in(2.37), by considering its expansion in powers of v. Recall identity (2.36) which statesthat!
N −1 YN
1
(1 + xi v) f (λ),
f (λ) = Fλ (x1 , . . . , xN ).
(HsHL f )(λ) =1 − sv
i=iDefining Hk by(1 − vs)

N −1

sHL

H

=

N
X

v k Hk ,(5.55)

k=0

we see that

Hk Fλ = ek (x1 , . . . , xN )Fλ .

(5.56)

The action of the Hk ’s on functions f (λ) can in principle be recovered using the vertex
weights w∗ in fig. 2.5 (without the denominator 1 − vs) which compose the sHL functions(1 − vs)N −1 F∗λ0 /µ0 . In the simplest cases k = 0 or N one can verify that
H0 = Id,

HN = Tµ1 · · · TµN ,where T is the shift operator(f (µ + ei ), if µi < µi−1 or i = 1;
(Tµi f )(µ) =0,otherwise.

Indeed, H0 requires no vertical arrows to change from µ to λ, and HN corresponds toadding a full horizontal path starting at zero and ending at N , so that the arrow configuration corresponding to λ is obtained from the one for µ by adding one vertical arrow at
location N .
When 1 ≤ k ≤ N − 1, explicit formulas for Hk look significantly more involved. We
need only the one for k = 1, and will not discuss the other operators Hk .2 In the next
statement, by agreement, we set µ0 = +∞, µN +1 = −∞.
Proposition 5.3.1. We have
H1 = h0,0 Id +X

0≤k<`≤Nhk,` Tµk+1 · · · Tµ` ,withh0,0 = −s
2

N
−1
X

q µj −µj+1 ,

j=1hk,` = (1 − q µk −µk+1 )(−s)`−k−1 q µk+1 −µ` (1 − s2 q µ` −µ`+1 ).

Appropriate scaling limits of the higher operators Hk could potentially lead to higher order differential
operators commuting with the deformed quantum Toda Hamiltonian H2 introduced below in this section.
We leave this investigation to a future work.

158

Proof. Express the action of H1 as
H1 f (µ) =X

H(µ; λ)f (λ),µ≺λ

where the term H(µ; λ) corresponds to the weight of a row of vertices having a configuration λ at the top and µ at the bottom. Recall that we are using down-right directed
paths as in fig. 2.5, and all the individual vertex weights are multiplied by (1 − vs).
somewhere in the bulk comes with the weight v(1 −Observe that each vertexµi −µi+1q). Therefore, terms proportional to v can only come from configurations withor with a single sequence of horizontal arrows
no horizontal arrows. The first case, corresponding to λ = µ, provides us with the term
h0,0 . For the latter case, let k be the column of the leftmost vertex emanating an horizontal arrow, and let ` be the column where the single horizontal path stops. Such
configuration corresponds to a partition λ = µ + ek+1 + · · · + e` . Isolating the coefficientof v in the expansion of product of vertex weights, we recover hk,` .

5.3.2

Scaling of the Pieri operators

Introduce the differential operators acting in the variables u1 , . . . , uN :
H1 :=

N
X

∂ui ;

(5.57)

i=1

N
X
1X 2
H2 := −
S −2(j−i) euj −ui (S − ∂ui )(S + ∂uj ).
∂ui +
2 i=1
1≤i<j≤N(5.58)In the second operator, the product is understood in the usual way as(S − ∂ui )(S + ∂uj ) = S 2 Id + S(∂uj − ∂ui ) − ∂ui ∂uj .
For the next result we define the rescalingq = e−ε ,q λi =e−ui
S N +1−2i,s = −e−εS .

(5.59)

Proposition 5.3.2. Under the scaling (5.59), we have1
(H1 − N ) ,
ε→0 ε


1
1 2
3
H2 = − lim 2 H1 − N + HN − 2 HN +
.
ε→0 ε
2
2

H1 = lim

We remark that the combinations of the refined Pieri operators leading to H1 andH2 are the same as in the q-Whittaker case [GLO12, Proposition 2.1], and correspond to
the scaling of eigenvalues in the proof of Theorem 5.3.3 below. The scaling (5.59) of thevariables, however, is different.
159

Proof of Proposition 5.3.2. First, expand the shift operator as ε → 0. From (5.59) we seethat the increment by 1 in µi corresponds to the increment by log(q −1 ) = ε in the scaled
variable ui . Therefore,
ε2
Tµk = Id + ε∂uk + ∂u2k + o(ε2 ),2and hence
X̀
ε2 X
∂uα +
Tµk+1 · · · Tµ` = Id + ε∂uα ,uβ + o(ε2 ).
2α=k+1
k+1≤α,β≤`This implies that

N
X
1 2
3
H − 2HN + = −ε
∂ui + o(ε2 ).
2 N
2
i=1

Next we address the scaling of H1 . Set(
S −2(`−k) eu` −ukak,` =
0(5.60)

if 1 ≤ k ≤ ` ≤ N,
else.

Expanding the coefficients hk,` , we have N −1ε2 2 Xaj,j+1 + o(ε2 ),h0,0 = 1 − εS + S
2
j=1
and (recall that we assume k < `)hk,` =(ak+1,` − ak,` − ak+1,`+1 + ak,`+1 )+ εS − (` − k − 1)(ak+1,` − ak,` ) + (` − k + 1)(ak+1,`+1 − ak,`+1 )


ε2 2
2
2
+ S (` − k − 1) (ak+1,` − ak,` ) − (` − k + 1) (ak+1,`+1 − ak,`+1 ) + o(ε2 ).
2Together with the action of the shifts T , we see that hk,` Tk+1 · · · T` expands as(ak+1,` − ak,` − ak+1,`+1 + ak,`+1 ) Id
+ ε − S(` − k − 1)(ak+1,` − ak,` ) Id + S(` − k + 1)(ak+1,`+1 − ak,`+1 ) Id
+ (ak+1,` − ak,` − ak+1,`+1 + ak,`+1 )

X̀

∂uα

α=k+1




ε
+
S 2 (` − k − 1)2 (ak+1,` − ak,` ) Id − S 2 (` − k + 1)2 (ak+1,`+1 − ak,`+1 ) Id
2X
+ (ak+1,` − ak,` − ak+1,`+1 + ak,`+1 )∂u2α ,uβ2k+1≤α,β≤`+ 2S − (` − k − 1)(ak+1,` − ak,` ) + (` − k + 1)(ak+1,`+1 − ak,`+1 )
160

 X̀

α=k+1

∂uα



+ o(ε2 ).

P
To evaluate the summation 0≤k<`≤N hk,` Tµk+1 · · · Tµ` , we use the identities in Proposition C.3.1. We obtainX
h0,0 +hk,` Tµk+1 · · · Tµ`0≤k<`≤N

=N +ε

N
X
i=1

∂ui + ε2

(

)
N
X

1X 2
∂ −
S 2(i−j) euj −ui (S − ∂ui ) S + ∂uj+ o(ε2 ).
2 i=1 ui 1≤i<j≤N

Together with (5.60) this yields the proof.
For the next result we employ the spin Whittaker functions in the additive parameters
ui , where the multiplicative parameters LN = (LN,N , . . . , LN,1 ) are expressed through theui ’s as
LN,i = S N +1−2i eui .
(5.61)
Denote the spin Whittaker function fX (LN ) in the additive parameters by fXadd (u1 , . . . , uN ).
Here X = (X1 , . . . , XN ) are such that |Xi | < S for all i.
Theorem 5.3.3. The spin Whittaker functions fXadd (u1 , . . . , uN ) in the additive variables(5.61) are eigenfunctions of the differential operators H1 (5.57) and H2 (5.58). In particular, we have
H1 fXadd (u1 , . . . , uN ) = − (X1 + · · · + XN ) fXadd (u1 , . . . , uN ),1
H2 fXadd (u1 , . . . , uN ) = − X12 + · · · + XN2 fXadd (u1 , . . . , uN ).
2

Proof. This result is a combination of the refined Pieri rules (5.56) viewed as eigenrelations
for the spin q-Whittaker functions, and the convergence of the functions (Theorem 5.1.14)
and the operators (Proposition 5.3.2). More precisely, under the scaling xi = e−εXi forthe eigenvalues ek (x1 , . . . , xN ) we have1e1 (x1 , . . . , xN ) − N −−→ −X1 − · · · − XN ,
ε→0
ε

and


1
3
1 2
1
2
e
(x
,
.
.
.
,
x
)
−
N
+
e
(x
,
.
.
.
,
x
)
−
2e
(x
,
.
.
.
,
x
)
+
−
−
→
(X +· · · XN2 ).
1
1
N
N
1
N
N
1
N
ε2
2
2 ε→0 2 1
This leads to the desired results.

5.3.3

Reduction to the quantum Toda Hamiltonian

It is natural to call the second order differential operator H2 (5.58) the deformed quantum
Toda Hamiltonian. Namely, it is diagonal in the spin Whittaker functions which (formally)
reduce, as S → +∞, to the classical gln Whittaker functions (Section 5.1.8). Further
161

justification to this name comes from the fact that the operator H2 itself degenerates as
S → +∞ to the usual quantum Toda Hamiltonian1
H2Toda := −

N
X

2 i=1

∂u2i +N−1
X

eui+1 −ui .

(5.62)

i=1

Proposition 5.3.4. As S → +∞, the operator H2 (5.58) converges to the quantum Toda
Hamiltonian H2Toda (5.62).
Proof. The factors S −2(j−i) , 1 ≤ i ≤ j ≤ N , in the sum in (5.58) decay at least as fast as
S −2 as S → +∞. Therefore, the only surviving contribution in the limit S → +∞ comesfrom the terms with j = i + 1, for which we haveS −2 eui+1 −ui (S − ∂ui )(S + ∂ui+1 ) → eui+1 −ui ,This completes the proof.

162

S → +∞.

Appendix A
Special functions
A.1

q-deformed quantities

Along the course of the thesis we largely made use of q-deformed quantities, such as qPochhammer symbols and q-hypergeometric series. The reader might consider these as
fairly common and established notions, but, for the sake of completeness, we still like todedicate this appendix to recall their definitions.
Assuming q is a parameter in the interval [0, 1), we define the q-Pochhammer symbol

n−1
if n ∈ Z>0 ,(1 − z)(1 − zq) · · · (1 − zq ),(A.1)(z; q)n = 1,
if n = 0,(1 − zq n )−1 (1 − zq n−1 )−1 · · · (1 − zq)−1 , if n ∈ Z<0 ,for every meaningful z ∈ C. We also denote the product of multiple q-Pochhammer
symbols of the same order in the compact notation(z1 ; q)n · · · (zk ; q)n = (z1 , . . . , zk ; q)n .

(A.2)

When n is positive, the q-Pochhammer symbol (A.1) is a polynomial in z and it admits
the expansion nXk (k2) n(z; q)n =(−z) q
,
(A.3)
k
q
k=0
where we introduced the q-binomial n(q; q)n=.
k q (q; q)k (q; q)n−k

The q-binomial admits the combinatorial expansion
 
X
k+1
n
=
q kIk−( 2 ) ,k qI⊂{1,...,n}
|I|=k

163

(A.4)

(A.5)

with I = {i1 , . . . , ik } and kIk = i1 + · · · + ik .
When we let the integer n grow to +∞, we see that the product in the left hand sideof (A.1) is convergent and hence we can defineY(z; q)∞ =(1 − zq j ).
(A.6)
j≥0

An important result concerning q-Pochhammer symbols is the summation identity
X (a; q)kk≥0(q; q)kzk =(za; q)∞(z; q)∞

for a ∈ C, |z| < 1,

(A.7)

which can be found in [Ism05], Theorem 12.2.5. and it is usually called q-binomial theorem. A slightly more general version of summation (A.7) is the so called q-Gauss summation ([Ism05], Theorem 12.2.4)
X  c n (a, b; q)nn≥0

ab(c, q; q)n=(c/a, c/b; q)∞,(c, c/(ab); q)∞

for |c/(ab)| < 1,or b ∈ q Z<0 .

The q-hypergeometric series X(a1 , . . . , ar+1 ; q)k k
a1 , a2 , . . . , ar+1z ,q, z =
r+1 φrb1 , b 2 , · · · , b r(b1 , . . . , br , q; q)k

(A.8)

(A.9)

k≥0

is defined for generic parameters a1 , . . . , ar+1 ∈ C, b1 , . . . , br ∈ C \ q Z<0 and |z| < 1. In
the case when at least one of the aj is of the form q −k , for some non-negative integer k,
the q-hypergeometric series (A.9) becomes a finite sum and its definition holds also for
more general complex numbers z. The regularized terminating q-hypergeometric functionis also defined as
 −n
 X
r
n
−n
; q)k Yq , a1 , . . . , ark (q(aj ; q)k (q k bj ; q)n−k .
(A.10)
q, z =zr+1 φ̄r
b1 , b2 , · · · , br
(q;
q)
k j=1
k=0
In the thesis we also made use of functions υj , defined as
υj (z) =X kj z kk≥1

1 − qk.

(A.11)

They are related to the more classical q-polygamma function [Wei]
X

q n+θ
,
ψq (θ) = − log(1 − q) + log(q)1 − q n+θ
n≥0
since




1
log(z)
υ0 (z) =log(1 − q) + ψq
log(q)
log(q)
164(A.12)and1dυj (z) = υj+1 (z).
(A.13)dzz
The inverse of the infinite q-Pochhammer symbol (A.6) is often called q-exponential andthrough it one can define a q-deformed notion of the common Laplace transform. For agiven f ∈ `1 (Z) the function
fe(ζ) =

X
n∈Z

f (n)
(q n ζ; q)for ζ ∈ C \ q Z∞(A.14)

is the q-Laplace transform of f . As for the usual Laplace transform, the operation f 7→ fe
admits an inverse. This is discussed, for example, in [IS19] and we do not report the exact
form of the inverse q-Laplace transform as we do not explicitly make use of it during thisthesis.
The q-gamma and the q-beta functions areΓq (X) =A.1.1(q; q)∞(1 − q)1−X ,(q X ; q)∞

Bq (X, Y ) =Γq (X)Γq (Y ),Γq (X + Y )for X, Y > 0. (A.15)

q-beta binomial distribution

Recall the definition of the q-deformed beta-binomial distribution ϕq,µ,ν from [Pov13],[Cor14].
Definition A.1.1. For m ∈ Z≥0 , consider the following distribution on {0, 1, . . . , m}:ϕq,µ,ν (j | m) = µj

(ν/µ; q)j (µ; q)m−j(q; q)m,(ν; q)m(q; q)j (q; q)m−j

0 ≤ j ≤ m.

(A.16)

When m = +∞, extend the definition as
ϕq,µ,ν (j | ∞) = µj

(ν/µ; q)j (µ; q)∞,(q; q)j (ν; q)∞j ∈ Z≥0 .

(A.17)The distribution depends on q and two other parameters µ, ν.
When 0 ≤ µ ≤ 1 and ν ≤ µ, the weights ϕq,µ,ν (j | m) are nonnegative.1 They also
sum to one:mX
ϕq,µ,ν (j | m) = 1,m ∈ {0, 1, . . .} ∪ {+∞} .
j=0Particular cases of the q-deformed beta-binomial are the q-negative binomial distribution defined as
X ∼ qNB(b, p)=⇒

P(X = n) = pn(b; q)n (p; q)∞,(q; q)n (pb; q)∞1

for all n ∈ Z≥0 (A.18)

These conditions do not exhaust the full range of (q, µ, ν) for which the weights are nonnegative. See,e.g., [BP18a, Section 6.6.1] for additional families of parameters leading to nonnegative weights.

165

and the q-Poisson distribution
X ∼ q−Poi(p)=⇒P(X = n) =pn(p; q)∞ ,(q; q)nfor all n ∈ Z≥0

The q-hypergeometric distribution is c n (a, b; q) (c, c/(ab); q)n∞ψq,a,b,c (n) =.
ab (c, q; q)n (c/a, c/b; q)∞(A.19)(A.20)

The fact that the weights (A.20) sum to one over n ∈ Z≥0 follows from the q-Gauss sum(A.8).

A.2

Classical special functions

It is well-known that Γq (X) converges toR ∞Γ(X) as q → 1 uniformly for X > 0, whereΓ is the usual gamma function Γ(z) = 0 e−t tz−1 dt, z > 0 (e.g., see [And86]). Hence
Bq (X, Y ) → B(X, Y ) uniformly for X, Y > 0, where B is the beta function
Z 1
Γ(x)Γ(y)tx−1 (1 − t)y−1 dt,
x, y > 0.
(A.21)=
B(x, y) =Γ(x + y)0
The inverse gamma distribution Γ−1 (α) on (0, +∞) with a parameter α > 0 is
Γ−1 (α)[x] =x−1−α e−1/x.
Γ(α)(A.22)The beta distribution on (0, 1) with (real) parameters m, n > 0 has density
B(m, n)[x] =xm−1 (1 − x)n−1
B(n, m)for x ∈ (0, 1).

We also recall that a random variable with negative binomial distribution has probability
mass function
kr k+r−1
NB(r, p)[k] = p (1 − p),for k ∈ Z≥0k

and r > 0, 0 ≤ p ≤ 1. Sampling x in the interval (0, 1) with B(m, n + k) law, where k isa NB(r, p) independent random variable generates the negative beta binomial distribution
on (0, 1). It has the probability density
(1 − p)r xm−1 (1 − x)n−1r, n + mp(1 − x) ,(A.23)
NBB(r, p, m, n)[x] =2 F1n–B(n, m)where we used the Gauss hypergeometric function X(a)k (b)k z k
a, b
z
=
F
,
2 1
c –
(c)
k!
k
k≥0

(A.24)and (r)k = r(r + 1) · · · (r + k − 1) is the Pochhammer symbol. Note that the inverse
gamma, beta, and the negative beta binomial are continuous distributions, while thenegative binomial is a discrete distribution.
166

Appendix B
Yang-Baxter equations
Here we review the Yang-Baxter equations used throughout the thesis.

b 2) R-matrices
Uq (slB.1
B.1.1

Basic cases

Most of the Yang-Baxter equations we use can be traced to the following basic one:
Proposition B.1.1. Consider the vertex weights w, r defined respectively in Figure 2.3
and Figure B.1. Then we haveXru/v (i2 , i1 ; k2 , k1 ) wv,s (i3 , k1 ; k3 , j1 ) wu,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3

=

X

(B.1)

wv,s (k30 , i1 ; j3 , k10 ) wu,s (i3 , i2 ; k30 , k20 ) ru/v (k20 , k10 ; j2 , j1 )k10 ,k20 ,k30

for all i1 , i2 , j1 , j2 ∈ {0, 1} and i3 , j3 ∈ Z≥0 . A visual representation of this equation is
given in Figure B.2.
j1

i2

i1

j2

rz (i1 , j1 ; i2 , j2 )1z(1−q)
1−qz

1−z
1−qz

q(1−z)
1−qz

1−q
1−qz

1

Figure B.1: In the top row we see all acceptable configurations of paths entering andexiting a vertex; below we reported the corresponding vertex weights rz (i1 , j1 ; i2 , j2 ).
Proof of Proposition B.1.1. This is established by a straightforward verification. Equation (B.1) appeared in several other works, including [Man14], [Bor17], [BW17].

167

wu,s
ru/v

j3

j3

k2

i1

wv,s

j2
=

k3

i2

j1

k1

k1

i1

j2

k3

i2

i3

j1

k2

i3

wv,s

ru/v

wu,s

Figure B.2: A schematic representation of the Yang-Baxter equation (B.1).
As explained in Section 2.2.2, from vertex weights wu,s one can define the dual weights∗
wv,s by changing u to 1/v, swapping the value of horizontal occupation numbers 0 ↔ 1,

and multiplying by (s − v)/(1 − sv) in order to assign weight 1 to the empty configuration.
These manipulations clearly preserve the structure of the Yang-Baxter equation, provided
that the same swapping of the occupation numbers is applied to the cross weight rz . This
leads to the definition of the cross weight Rz , see Figure B.3, also normalized so that theempty configuration has weight 1.
j1

i2

i1

j2

Rz (i1 , j1 ; i2 , j2 )

1

1−qz
1−z

(1−q)z
1−z

1−qz
1−z

1−q
1−z

q

Figure B.3: The cross vertex weights Rz (i1 , j1 ; i2 , j2 ).
Proposition B.1.2. Consider the vertex weights w, w∗ and R, defined respectively in
Figures 2.3 and 2.5, and Figure B.3. Then we haveX∗
Ruv (i2 , i1 ; k2 , k1 ) wv,s(i3 , k1 ; k3 , j1 ) wu,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3

=

X

∗
wv,s
(k30 , i1 ; j3 , k10 ) wu,s (i3 , i2 ; k30 , k20 ) Ruv (k20 , k10 ; j2 , j1 )(B.2)

k10 ,k20 ,k30

for all i1 , i2 , j1 , j2 ∈ {0, 1} and i3 , j3 ∈ Z≥0 .

B.1.2

FusionThrough a fusion procedure we generalize vertex weights wu,s and allow configurations
with multiple paths crossing a vertex in the horizontal direction. This technique of generalizing solutions to the Yang-Baxter equation was originally introduced in [KRS81] and168

consist in collapsing together a series of vertically attached vertices with spectral parameters forming a geometric progression of ratio q. The fusion of vertex weights also admits
a probabilistic interpretation [CP16], [BP18a], [BW17].
Define the fused vertex weight
1(−1)i1 +j2 q 2 i1 (i1 −1+2j1 ) sj2 −i1 ui1 (u/s; q)j1 −i2 (q; q)j1
(J)
wu,s (i1 , j1 ; i2 , j2 ) = 1i1 +j1 =i2 +j2(q; q)i1 (q; q)j2 (su; q)j1 +i1!
−i1 −i2J× 4 φ3q;q, suq , qs/u2

1+j2 −i1(B.3)

q, q ,s ,q,q

1−i2 −j2 +J

where 4 φ3 is the regularized q-hypergeometric series (A.9). Here J is originally a positive
integer representing the number of vertices which were fused together. However, it is easy
to see that w(J) depends on q J in a rational way, thus q J can be regarded as the fourth
independent parameter in (B.3) (along with u, s, and q). Since the regularized series 4 φ3
terminates, (B.3) depends on all these parameters in a rational way. Moreover, in case
i1 , i2 → ∞, the weight w loses its dependence of j1 and we have(J)
lim wu,s(n, j1 ; n + j1 − j2 , j2 ) = (−uq J )j2n→∞(q −J ; q)j2 (suq J ; q)∞.
(q; q)j2 (su; q)∞(B.4)

Just as in the J = 1 case, the fused boundary weight is obtained removing the normalization factor from (B.4), and we define
(J)
wu,s



∞
∞

k



= (−uq J )k(q −J ; q)k.
(q; q)k

(B.5)

This normalization is needed to assign weight 1 to the empty configuration of paths in
the grid. The fused analog of the dual weights w is defined similarly to (2.9):
∗,(I)
wv,s
(i1 , j1 ; i2 , j2 ) =(s2 ; q)i1 (q; q)i2 (I)w (i2 , j1 ; i1 ; j2 ).
(q; q)i1 (s2 ; q)i2 v,s(B.6)

These quantities also depend on v, s, q, and q I in a rational way.
What makes the fused weights remarkable is that they satisfy a general version of
the Yang-Baxter equation (previously in Section B.1.1 the horizontal occupation numbers
had to be either 0 or 1). In order to state this equation we need to consider the fusion of
the cross weights Rz leading to1

q i2 i1 + 2 j2 (j2 −1)+j2 J (−z)j2 (q; q)j1
(z; q)j1 +i2 (q; q)j2 (q; q)i2 (q 1−J /z; q)i1 −j1!
q −i2 ; q −i1 , zq I , q 1−J /z× 4 φ3q, q .
−J
1+j2 −i2 1−i1 −j2 +I
q ,q,qRz(I,J) (i1 , j1 ; i2 , j2 ) := 1i2 +j1 =i1 +j2

169

(B.7)

Proposition B.1.3. Consider the weights w(J) , w∗,(I) and R(I,J) defined in (B.3), (B.6),(B.7). Then we have
X
(I,J)
∗,(I)
(J)
Ruv
(i2 , i1 ; k2 , k1 ) wv,s(i3 , k1 ; k3 , j1 ) wu,s(k3 , k2 ; j3 , j2 )k1 ,k2 ,k3=

X∗,(I) 0(J)(I,J) 0wv,s(k3 , i1 ; j3 , k10 ) wu,s(i3 , i2 ; k30 , k20 ) Ruv(k2 , k10 ; j2 , j1 ),(B.8)

k10 ,k20 ,k30

for all admissible values of i1 , i2 , j1 , j2 (that is, i1 , j1 ∈ {0, 1, . . . , I − 1} for I a positive
integer, or i1 , j1 ∈ Z≥0 if q I is generic, and similarly for i2 , j2 ), and i3 , j3 ∈ Z≥0 .
Note that in (B.8) (and in all other Yang-Baxter equations in this Appendix) for fixedboundary occupation numbers i1 , i2 , i3 , j1 , j2 , j3 the sums over k and k 0 indeces in bothsides are finite due to arrow preservation, so there are no convergence issues when i3 andj3 are finite.
Remark B.1.4. The fused cross weights R(I,J) inherit symmetries of the unfused weight
R of Figure B.3. One of these is given by the identity
Rz(I,J) (i1 , j1 ; i2 , j2 ) = Rz(J,I) (j1 , i1 ; j2 , i2 )for all i1 , j1 , i2 , j2 ∈ Z≥0 .

(I,J)

Proposition B.1.5. Consider the vertex weight Rz
X

defined in (B.7). Then we haveRz(I,J) (a2 , a1 ; k1 , k2 ) = Rz(I,J) (0, I; 0, I) = Rz(I,J) (J, 0; J, 0) =k1 ,k2(B.9)(zq I ; q)∞ (zq J ; q)∞(z; q)∞ (zq I+J ; q)∞(B.10)

for all a1 , a2 ∈ Z≥0 .
Proof. The second and the third equalities in (B.10) follow, after algebraic manipulations,(I,J)
from the definition of the fused cross weight Rz
given in (B.7).
The first equality in (B.10) is a trivial check in the case when I = J = 1, using thedefinition of Rz of Figure B.3. It lifts to more general I, J as the fusion procedure does
not affect the structure of the identity.

B.1.3

Spin q-Whittaker specialization

The spin q-Whittaker specialization of the general fused weights (B.3), (B.6) is obtained by setting u = s and q J = −x/s (recall that one can regard q J as a generic(J)parameter). After this specialization the complicated expression wu,s (i1 , j1 ; i2 , j2 ) (B.3)
factorizes and becomes Wx,s (i1 , j1 ; i2 , j2 ) given by (2.13). Analogously, the dual fused∗,(I)∗weight wv,s (i1 , j1 ; i2 , j2 ) (B.6) turns into Wy,s(i1 , j1 ; i2 , j2 ) (2.29) after setting v = s andI
q = −y/s.
The most general Yang Baxter equation (B.8) specializes to Yang-Baxter equations∗
involving Wx,s and Wy,sas long as the corresponding specializations are applied to the(I,J)
cross weight Ruv , too. Let us record the resulting identities:
170

Proposition B.1.6. We have the following Yang-Baxter equations:X∗
Rx,v,s (i2 , i1 ; k2 , k1 ) wv,s(i3 , k1 ; k3 , j1 ) Wx,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3=XX

k10 ,k20 ,k30

(B.11)∗
wv,s(k30 , i1 ; j3 , k10 ) Wx,s (i3 , i2 ; k30 , k20 ) Rx,v,s (k20 , k10 ; j2 , j1 );∗Rx,y,s (i2 , i1 ; k2 , k1 ) Wy,s(i3 , k1 ; k3 , j1 ) Wx,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3=

X(B.12)∗
Wy,s(k30 , i1 ; j3 , k10 ) Wx,s (i3 , i2 ; k30 , k20 ) Rx,y,s (k20 , k10 ; j2 , j1 ).

k10 ,k20 ,k30

The cross vertex weights in (B.11) are given in Figure B.4. In the third identity (B.12)
the cross vertex weights do not factorize (here i1 , i2 , j1 , j2 ∈ Z≥0 ):1

q i2 i1 + 2 j2 (j2 −1) (sx)j2 (q; q)j1
Rx,y,s (i1 , j1 ; i2 , j2 ) = 1i2 +j1 =i1 +j2 2(s ; q)j1 +i2 (q; q)j2 (q; q)i2 (−q/(sx); q)i1 −j1
q −i2 ; q −i1 , −sy, −q/(sx)× 4 φ3 q, q  .
1+j2 −i21−i1 −j2−s/x, q
, −yq
/s
j1

i2

i1

j2

g

g

Rξ,v,s (i1 , j1 ; i2 , j2 )gg

1−q sv1−svgg

ξv+q sv
1−sv

g

g−1

g+1

(B.13)

g
g

1−q
1−sv

q g +vξ
1−sv

Figure B.4: The cross vertex weights Rx,v,s (i1 , j1 ; i2 , j2 ), j1 , j2 ∈ {0, 1}, i1 , i2 ∈ Z≥0 .
So far we have listed Yang-Baxter relations between weights W and their dual form
W . Nevertheless eq. (B.1) extends, after fusion also to a Yang-Baxter equation for pairsof W weights. Let us introduce the cross vertex weight∗r̄x,y (i1 , j1 ; i2 , j2 ) := 1i1 +j1 =i2 +j2 1i1 ≥j2 (y/x)j2(−s/y; q)j2 (y/x; q)i1 −j2 (q; q)i1.
(q; q)j2 (q; q)i1 −j2 (−s/x; q)i1

(B.14)

Proposition B.1.7. For any i1 , i2 , i3 , j1 , j2 , j3 ∈ Z≥0 , we haveXr̄x,y (i2 , i1 ; k2 , k1 )Wy,s (i3 , k1 ; k3 , j1 )Wx,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3 ≥0=XWx,s (i3 , i2 ; k30 , k20 )Wy,s (k30 , i1 ; j3 , k10 )r̄x,y (k20 , k10 ; j2 , j1 ),k10 ,k20 ,k30 ≥0(B.15)

where W are the bulk weights defined by (2.13). See Figure B.2 for a graphical interpretation.
171

Proof. This is obtained in [BW17, Corollary 4.3] via fusion from the elementary Yang–
Baxter equation for the higher spin sl2 vertex model. Note that the claim of [BW17,
Corollary 4.3] contains a typo: the spectral parameters x, y in the definition of the crossvertex weight should be swapped. This is corrected here by defining Rx,y in (B.14) withparameters already swapped.

j3

j3
i1

k2

j2
=

k3

i2

j1

k1

k10

i1

j2

i1

k2

k30

j2

k20

j1

i2

j1

k1

i3

i1
=

k3

i2

i3

j3

j3

i2

j2

k20

j1

i3

i3

(a)

k10
k30

(b)

Figure B.5: Graphical representation of the Yang–Baxter equations with non-dual weights
and dual weights resp. (a) and (b).

B.1.4

Scaled geometric specialization(J)

The scaled geometric specialization of the general fused weight wu,s is given by settingu = −α, q J = 1/ and taking the limit  → 0. Analogously we can specialize the dual∗,(I)weight wv,s taking v = −β, q I = 1/ and again  → 0. In this case the expessions (B.3),(B.5) simplify:
−i1 −i2
i1
j2
1+i
+j
q ; q , −sα(−α/s) (−s) (q; q)j1
sq 2 2 w̆α,s (i1 , j1 ; i2 , j2 ) = 1i1 +j1 =i2 +j2
,
q, −
3 φ2
(q; q)i1 (q; q)j2αs2 , q 1+j2 −i1(B.16)
and


∞
αk
=
.
(B.17)w̆α,sk(q; q)k
∞∗
The dual weights w̆β,sare defined in the usual way as in (2.9).
j1

i2

i1

j2

(sHL,sg)
Rα,v(i1 , j1 ; i2 , j2 )

g

g

g

g

αv

1

(sHL,sg)

Figure B.6: The cross vertex weight Rα,v

172

g

g−1

g+1

g

1 − qgq g + vα(i1 , j1 ; i2 , j2 ), j1 , j2 ∈ {0, 1}, i1 , i2 ∈ Z≥0 .

We also consider the scaled geometric specialization of the fused cross weight R(I,J) ,in this case in the parameters v, q I , defining
−i1 −i2
1+i
+j
J
i2 −J
q ; q , −uβq 1 2
(−uq β) (q ; q)i2
(sg,J)
q,−
φ.
Ru,β (i1 , j1 ; i2 , j2 ) = 1i2 +j1 =i1 +j2
3
2(q; q)i2 (q −J ; q)i1
uq J β−J
1+j2 −i2
q ,q(B.18)The scaled geometric specialization of R(I,J) in the parameters u, q J can be derived from
(B.18) using the symmetry (B.9) and it is(I,sg)(sg,I)
Rα,v(i1 , j1 ; i2 , j2 ) = Rv,α(j1 , i1 ; j2 , i2 ).

(B.19)(I,sg)

Further degenerations of Rα,v involve specializations of parameters v, q I in one of thethree cases, sHL(v) (which is simply I = 1), sqW(y), or sg(β). These cross vertex weights
are given, respectively, in Figure B.6 and below:−j1 −j2
1+j+i
j2
q ; q , −sα q 1 2(αy) (−s/y; q)j2
(sqW,sg)

,
q,
Rα,y
(i1 , j1 ; i2 , j2 ) = 1i2 +j1 =i1 +j2
3 φ2(q; q)j2 (−s/y; q)j1
αy1+i1 −j1−s/y, q(B.20)!
−j1 −j2
q ;q
q 1+j1 +i2(αβ)j2(sg,sg)q,.
(B.21)
Rα,β (i1 , j1 ; i2 , j2 ) = 1i2 +j1 =i1 +j2
2 φ1(q; q)j2
αβ
q 1+i1 −j1These cross vertex weights enter a number of Yang-Baxter equations which are specializations of the general fused one (B.8):
Proposition B.1.8. We have the following Yang-Baxter equations:X(sHL,sg)∗
Rα,v(i2 , i1 ; k2 , k1 ) wv,s(i3 , k1 ; k3 , j1 ) w̆α,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3

X=
X(B.22)

k10 ,k20 ,k30
(sqW,sg)∗Rα,y(i2 , i1 ; k2 , k1 ) Wy,s(i3 , k1 ; k3 , j1 ) w̆α,s (k3 , k2 ; j3 , j2 )k1 ,k2 ,k3=X∗(sHL,sg) 0wv,s(k30 , i1 ; j3 , k10 ) w̆α,s (i3 , i2 ; k30 , k20 ) Rα,v
(k2 , k10 ; j2 , j1 );X∗(sqW,sg) 0Wy,s(k30 , i1 ; j3 , k10 ) w̆α,s (i3 , i2 ; k30 , k20 ) Rα,y(k2 , k10 ; j2 , j1 );(B.23)

k10 ,k20 ,k30
(sg,sg)Rα,βk1 ,k2 ,k3=∗(i2 , i1 ; k2 , k1 ) w̆β,s(i3 , k1 ; k3 , j1 ) w̆α,s (k3 , k2 ; j3 , j2 )X(sg,sg)∗w̆β,s(k30 , i1 ; j3 , k10 ) w̆α,s (i3 , i2 ; k30 , k20 ) Rα,β(k20 , k10 ; j2 , j1 ).

(B.24)

k10 ,k20 ,k30

Dual cases of (B.22),(B.23),(B.24) obtained swapping the specializations are easily derived
making use of the symmetry of the cross weight (B.19).
173

In Chapter 2, Cauchy Identities for spin Hall-Littlewood and spin q-Whittaker functions were stated as corollaries of the Yang-Baxter equations given in this appendix. In
particular, the emergence of the prefactors in the right-hand sides of all the skew Cauchy
identities can be traced to Proposition B.1.5.

B.2

Corner R-matrices

In this section we list the Yang–Baxter equations involving the class of vertex weight W .

B.2.1

Non dual sqW/sqW Yang–Baxter equation

Proposition B.2.1. For any i1 , i2 , j1 , j2 ∈ Z≥0 , we haveXr̄x,y (i2 , i1 ; k2 , k1 )Wy,s (k1 )Wx,s (k1 , k2 ; j2 , j1 )Wx,s (j1 )k1 ,k2 ≥0(B.25)= Wx,s (i2 )Wy,s (i2 , i1 ; j2 , j1 )Wy,s (j1 ),where W are the right corner weight defined by (2.14). See Figure B.5 (a) for an illustration.
Proof. Expanding both right and left-hand side of (B.25) and simplifying common factors
we end up with the identityi2X(y/x)k−j1

k=j1(−sy; q)i2 −j1(y/x; q)i2 −k (−sx; q)k−j1=,(q; q)i2 −k (q; q)k−j1(q; q)i2 −j1

which follows from the q-Gauss summation (A.8).

B.2.2

Yang–Baxter equations with dual weights∗
Our additional Yang–Baxter equations involve the dual sHL weights wv,swhich are givenin (2.9) in the text and the dual sqW weights (2.28)–(2.30). We use the cross vertex
weights Rx,v,s given in Figure B.4 and the cross vertex weights (B.13).

Proposition B.2.2. For any i1 ∈ {0, 1} and i2 , i3 , j2 , j3 ∈ Z≥0 , we haveX
Rx,v,s (i2 , i1 ; k2 , k1 ) W ∗, (i3 , k1 ; k3 ) Wx,s (k3 , k2 ; j3 , j2 ) Wx,s (j2 )k1 ,k2 ,k3 ≥0=X∗Wx,s (i3 , i2 ; k30 , k20 ) Wx,s (k20 ) wv,s(k30 , i1 ; j3 , k10 ) W ∗, (k20 , k10 ; j2 ).

k10 ,k20 ,k30 ≥0

See Figure B.5 (b) for a graphical interpretation.

174(B.26)

j2
i1

j1

j2

k2

i1

j1

=

k1

i2

j1

j3
i1

j1

i2
j2

k30

i2

k1

k20
k20

i3

i3

(a)

j2
k10

i1
=

k3

i2

k1

j2

k2

i2

j3

j2

(b)

Figure B.7: Graphical representation of the Yang–Baxter equations with corner Rmatrices.
Proof. Consider separately the cases i1 = 0 and i1 = 1. Start with i1 = 0. We see that
(B.26) is nontrivial only when i2 + i3 = j2 + j3 (say j2 = i2 + i3 − j3 ) and j3 ≥ i2 . Under
these conditions we have
Rx,v,s (i2 , 0; i2 , 0)Wx,s (i3 , i2 ; j3 , i2 + i3 − j3 )Wx,s (i2 + i3 − j3 )+ Rx,v,s (i2 , 0; i2 + 1, 1)Wx,s (i3 − 1, i2 + 1; j3 , i2 + i3 − j3 )Wx,s (i2 + i3 − j3 )∗= wv,s(j3 , 0; j3 , 0)Wx,s (i3 , i2 ; j3 , i2 + i3 − j3 )Wx,s (i2 + i3 − j3 )∗+ wv,s(j3 − 1, 0; j3 , 1)Wx,s (i3 , i2 ; j3 − 1, i2 + i3 − j3 + 1)Wx,s (i2 + i3 − j3 + 1).

After the required simplifications, the previous relation reduces to(1 − q i2 sv)(1 + sxq j3 −i2 −1 ) + (xv + svq i2 )(1 − q j3 −i2 )= (1 − svq j3 )(1 + sxq j3 −i2 −1 ) + xv(1 − q j3 −i2 )(1 − s2 q j3 −1 ),that can be checked directly.
When i1 = 1, as in the i1 = 0 case, (B.26) is an equality between sums of at most two
terms that after simplification reduces to(1 − q i2 )(1 + sxq j3 −i2 ) + (q i2 + xv)(1 − q j3 −i2 +1 )= (1 − q j3 +1 )(1 + sxq j3 −i2 ) + x(1 − q j3 −i2 +1 )(v − sq j3 ),which is again checked directly.
Proposition B.2.3. For any i1 , i2 , i3 , j1 , j2 , j3 ∈ Z≥0 , we haveX
Rx,y,s (i2 , i1 ; k2 , k1 )W ∗, (i3 , k1 ; k3 )Wx,s (k3 , k2 ; j3 , j2 )Wx,s (j2 )k1 ,k2 ,k3 ≥0=(−sy; q)∞ X(s2 ; q)∞ 0 0 0∗
Wx,s (i3 , i2 ; k30 , k20 )Wx,s (k20 )Wy,s(k30 , i1 ; j3 , k10 )W ∗, (k20 , k10 ; j2 ).

k1 ,k2 ,k3 ≥0(B.27)

Proof. This follows from the analogous relation (B.26). In fact both the R-matrix R andthe vertex weight W ∗ can be constructed respectively from R and w∗ via fusion with175

respect to the spectral parameter v (see [BW17], [BMP19] for details). The coefficient(−sy;q)∞arises from fusion of w∗ and does not simplify since in the left hand side of (B.26)(s2 ;q)∞the bulk weight w∗ is missing. One can check that the fusion procedure preserves identity(B.26) and hence (B.27) holds.

B.3

Nonnegativity of terms in the Yang-Baxter equations

Here we list conditions which are sufficient for the nonnegativity of all terms in both sidesof the Yang-Baxter equations described in the previous parts of this Appendix. We will
not discuss which of these assumptions are necessary. If the terms are nonnegative, then astochastic bijectivization of the Yang-Baxter equation exists. We assume that s ∈ (−1, 0)
and q ∈ (0, 1) throughout the rest of the subsection.
∗
First, the weights wu,s and wv,sgiven in Figure 2.3 and Figure 2.5 are nonnegative
for u, v ∈ [0, 1]. The cross vertex weights ru/v from Figure B.1 are nonnegative when in
addition u < v. Thus,
All summands in both sides of the Yang-Baxter equation (B.1) containing the weights wu,s , wv,s , and ru/v are nonnegative if 0 ≤ u < v ≤ 1.
Next, the cross vertex weights Ruv from Figure B.3 are nonnegative when 0 ≤ uv < 1.
Therefore,
All summands in both sides of the Yang-Baxter equation (B.2) con∗taining the weights wu,s , wv,s, and Ruv are nonnegative if u, v ∈ [0, 1).
This in fact implies that (u, v) ∈ Adm for the sHL/sHL skew Cauchy
structure (definition 3.1.4).
∗
are
Let us now turn to the spin q-Whittaker weights. The weights Wx,s and Wy,s−1nonnegative when x, y ∈ [−s, −s ]. The weights Rx,v,s from Figure B.4 are nonnegative
when v ∈ [0, 1) and x ∈ [−s, s−1 ]. Thus, we have

All summands in both sides of the Yang-Baxter equation (B.11) con∗taining the weights Wx,s , wv,s, and Rx,v,s are nonnegative if v ∈ [0, 1)−1
and x ∈ [−s, −s ].
∗Further, let us consider (B.12) containing Wx,s , Wy,s
, and the non-factorized weights
Rx,y,s (B.13). Their nonnegativity is not as straightforward, and requires an additional
restriction on the parameters s and q:√
Proposition B.3.1. For x, y ∈ [−s, −s−1 ], q ∈ (0, 1) and s ∈ [− q, 0), we haveRx,y,s (i1 , j1 ; i2 , j2 ) ≥ 0

for all i1 , j1 , i2 , j2 ∈ Z≥0 .

The proof of this proposition is similar to [CMP19, Proposition 3.1], with an additional
simplification in the second case due to a symmetry of Rx,y,s .
176

Proof of Proposition B.3.1. Throughout the proof we will assume that i2 + j1 = i1 + j2 .
We need to show that −iqsj2 (−yq 1−i1 −j2 /s; q)i2
q 2 , − sx
, −sy, q −i1
q, q ≥ 0.
(B.28)
4 φ3− ys q 1−i1 −j2 , − xs , q 1+j2 −i2(−q/(sx); q)i1 −j1Here we used (A.9) to get to the usual q-hypergeometric function, and also the fact thatthe remaining prefactor in Rx,y,s having the form1

q i2 i1 + 2 j2 (j2 −1) xj2 (q; q)j1 (−s/x; q)i2 (q 1+j2 −i2 ; q)i2(s2 ; q)j1 +i2 (q; q)j2 (q; q)i2
is nonnegative under our parameter restrictions in a straightforward way.
We will use Watson’s transformation formula [GR04, (III.19)] −n(d/b; q)n (d/c; q)nq , a, b, c
q, q =
8 φ7
4 φ3d, e, f –(d; q)n (d/(bc); q)nq −n , σ, qσ 1/2 , −qσ 1/2 , fa , ae , b, c
, ef , efaq
σ 1/2 , −σ 1/2 , e, f, ef
ab ac

n

ef q nq,bc(B.29)

where def = abcq 1−n and σ = ef /aq.
Case 1. When i2 ≤ j2 , we apply (B.29) to (B.28) with n = i2 . The prefactor in (B.29)
combined with the one from (B.28) becomes(q 1−i1 −j2 /s2 ; q)i2 (−yq 1−j2 /s; q)i2
sj2.
(−q/(sx); q)i1 −j1(q 1−j2 /s2 ; q)i2
We havei2
Y(q 1−i1 −j2 /s2 ; q)i2
q m−j2 q −i1 − s2
≥ 0,=
(q 1−j2 /s2 ; q)i2
q m−j2 − s2
m=1

since m − j2 ≤ 0 in the product. We also havej2 −i2 i2 Ysj2 (−yq 1−j2 /s; q)i2
y k−j2  Y
q 1−m
j2
=s
≥ 0,
1+ q
1+
(−q/(sx); q)i1 −j1
s
sx
m=1
k=1

since all factors above (including sj2 ) are nonpositive, and there is a total of 2j2 of them.
The q-hypergeometric function after applying (B.29) to (B.28) takes the form2
q −i2 , σ, qσ 1/2 , −qσ 1/2 , −sxq j2 −i2 , sq , −sy, q −i1
i1 +j2 +1
q

,
q,
8 φ7
s
s j2 −i2 2 j1 2 j2
xy
1/2
1/2
1+j2 −i2
σ , −σ , − , q,− q,s q ,s qxywith σ = s2 q j2 −i2 −1 ∈ (0, 1) because s2 ≤ q. One readily sees that each summand in this(terminating) q-hypergeometric series is nonnegative. Indeed, the only negative signs may
come from (q −i2 ; q)k , (q −i1 ; q)k , and (s2 q −1 ; q)k . However, the product of the former two
factors is always nonnegative, and (s2 q −1 ; q)k ≥ 0 also due to our additional parameter
restriction s2 ≤ q. This implies the nonnegativity of Rx,y,s (i1 , j1 ; i2 , j2 ) when i2 ≤ j2 .
177

!

,

Case 2. When i2 > j2 , the claim follows due to the symmetry of Rx,y,s . Namely, bymeans of Remark B.1.4, we haveRx,y,s (i1 , j1 ; i2 , j2 ) = Ry,x,s (j1 , i1 ; j2 , i2 )for all i1 , j1 , i2 , j2 ∈ Z≥0 . This completes the proof.
Proposition B.3.1 implies thatAll summands in both sides of the Yang-Baxter equation (B.12) con∗, and Rx,y,s are nonnegative if x, y ∈taining the weights Wx,s , Wy,s√−1[−s, −s ], q ∈ (0, 1), and s ∈ [− q, 0).
Finally, we address the nonnegativity of terms of the Yang-Baxter equations involving
scaled geometric specializations from Proposition B.1.8.
Proposition B.3.2. For α ∈ [0, −s−1 ], q ∈ (0, 1) and s ∈ (−1, 0) we havefor all i1 , j1 , i2 , j2 ∈ Z≥0 .

w̆α,s (i1 , j1 ; i2 , j2 ) ≥ 0
Proof. Under our assumptions the prefactor(−α/s)i1 (−s)j2 (q; q)j1
(q; q)i1 (q; q)j2
is nonnegative. To check the remaining term, we write down the generic summand of theterminating q-hypergeometric series as (cf. (A.9)):k −i1(q ; q)k −i2−sq 1+j2 +i2(q ; q)k (−sα; q)k (s2 q k ; q)i1 −k (q 1+j2 −i1 +k ; q)i1 −k ,α(q; q)kwhere k = 0, . . . , i1 . The leading monomial term, along with (s2 q k ; q)i1 −k and (q; q)k are always nonnegative. The q-Pochhammer symbols of q −i1 and q −i2 either vanish, or they bothcarry a sign (−1)k , so that their contribution is nonnegative too. Next, (q 1+j2 −i1 +k ; q)i1 −kis either nonnegative if 1 + j2 − i1 + k > 0, or vanishes if 1 + j2 − i1 + k ≤ 0 (in thelatter case, the last term of the product has power j2 ≥ 0, which means that that productpasses through 1 − q 0 = 0). Finally, (−sα; q)k ≥ 0 because α ≤ −s−1 .
(sHL,sg)

Proposition B.3.2 and the explicit form of Rα,v

(Figure B.6) implies thatAll summands in both sides of the Yang-Baxter equation (B.22) contain(sHL,sg)∗
ing the weights w̆α,s , wv,s, and Rα,v
are nonnegative if α ∈ [0, −s−1 ],v ∈ [0, 1).
In order to demonstrate the nonnegativity of (B.23) we consider the corresponding crossvertex weight:
Proposition B.3.3. For α ∈ [0, −s−1 ] and y ∈ [−s, −s−1 ], we havefor all i1 , j1 , i2 , j2 ∈ Z≥0 .

(sqW,sg)Rα,y(i1 , j1 ; i2 , j2 ) ≥ 0

178

Proof. Assume first that y > −s. In (B.20), the factors outside 3 φ2 are nonnegative. In
the expansion of 3 φ2 using (A.9), one readily sees that all terms are nonnegative similarly
to the proof of Proposition B.3.2 above (here we use the fact that −sα and −s/y are lessthan 1 because of our assumptions).
We can now take the limit y → −s and show that the weight R(sqW,sg) survives thistransition. To do so, expand 3 φ2 using (A.9), and collect terms containing −s/y:(−s/y; q)j2 (−q k s/y; q)j1 −k= (−q k s/y; q)j2 −k ,(−s/y; q)j1
with k = 0, . . . , min(j1 , j2 ). The last expression is nonsingular at y = −s, and is nonnegative.
Therefore,
All summands in both sides of the Yang-Baxter equation (B.23) con(sqW,sg)∗
taining w̆α,s , Wy,s
, and Rα,yare nonnegative if α ∈ [0, −s−1 ],−1y ∈ [−s, −s ].
We come now to the last Yang-Baxter equation we stated (B.24), in which one readily(sg,sg)sees (similarly to Propositions B.3.2 and B.3.3 above) that Rα,β is nonnegative when0 ≤ α, β ≤ −s−1 . Therefore,
All summands in both sides of the Yang-Baxter equation (B.24) con(sg,sg)∗
taining w̆α,s , w̆β,s, and Rα,β are nonnegative if α, β ∈ [0, −s−1 ].

179Appendix C
ExtrasC.1

Bounds for φl , ψl , Φx, Ψx

We collect here some useful bounds for the quantities φl , ψl , Φx , Ψx defined in eqs. (4.8)
to (4.11). Terms Φx , Ψx can be further decomposed as(2)Φx (n) = Φ(1)x (n) + Φx (n),(2)Ψx (n) = Ψ(1)x (n) + Ψx (n),obtained separating from the integration (4.10) (resp. (4.11)) the contribution of polew = d (resp. z = v) from that of other poles. Their exact expression was given ineqs. (4.22) to (4.25).
Proposition C.1.1. Let v < d . Then, for all fixed x, there exist constants Γ1 , Γ2 , Γ3 , Γ4 >0, such that
−Γ2 |n||φl (n)|, |ψl (n)|, |Φx (n)|, |Φ(2)x (n)| < Γ1 eand(
Γ1 e−Γ2 |n||Ψx (n)| <Γ3 e−Γ4 |n|

for all n ∈ Z

if n ∈ Z<0
if n ∈ Z≥0 .

(C.1)

(C.2)

Moreover Γ1 , Γ2 can be chosen so that their relative bounds also hold for v, d in the region(4.16) (in this case the parameter b appearing in the definition of τ (4.12) satisfies qv <b < d ).
(2)

Proof. We start with the terms φl , Φx (n), Φx . Evaluating the complex integrals as sumsof residues it is straightforward to get the inequalities


1n≥0
|n|
(2)
≤ Γ1 e−Γ2 |n| ,+ 1n<0 max(|ξi si |)|φl (n)|, |Φx (n)|, |Φx (n)| ≤ const.τ (n)i≥2|d ||n|for some constants Γ1 , Γ2 depending on the integrand functions but not on n.
180

To obtain a similar bound for the term ψl (n) we distinguish two cases. When n is
positive we take the contour C to be a circle of radius r+ so that qv < r+ < b. On theother hand, when n is negative we take C to be a circle of radius r− strictly bigger thanc, not containing any of the numbers ξi /si (we remark that the definition itself of τ (n)and of numbers b, c is tailor-made for these conditions to be possible). With this choices
we easily get!
11
1n≥0 r+|n| + 1n<0 |n| ≤ Γ1 e−Γ2 |n| .
|ψl (n)| ≤ const.
τ (n)r−An argument equivalent to that used for ψl can be carried to show (C.2). The only
difference here is that the radius r+ has to be chosen so that v < r+ < b and hence wecannot extend this bound to the region qv < b ≤ v.
Proposition C.1.2. Let v satisfy (4.4) and v < d or possibly (4.16). Then, for each x,
there exist constants Γ1 , Γ2 > 0 such that
−Γ2 |n||φl (n)Ψx (n)|, |Φ(2).
x (n)Ψx (n)| < Γ1 e

(C.3)

Proof. From Proposition C.1.1 we see that we only have to prove (C.3) for positive n’s.
When this is the case we see directly from the integral expression (4.11) and (4.8) that(2)
we can bound both |φl (n)Ψx (n)| and |Φx (n)Ψx (n)| with some quantity proportional to|v + |n,mink≥2 |ξk sk |n(C.4)

by simply taking the C contour as a circle of radius v + , for  being sufficiently small.
Due to the conditionv < min |ξk sk |,k≥2

we see that  can be chosen so that (C.4) decays to zero and this completes the proof.
Proposition C.1.3. Let v, d satisfy (4.16). Then, for each fixed x, there exist constants
Γ1 , Γ2 > 0 such that(2)
−Γ2 |n||f (n)Φ(1).
x (n)Ψx (n)| < Γ1 e
Proof. We use the integral expression (4.25). When n is positive we take the integration
contour C1 to be a circle of radius qv + . A bound we can easily obtain is(2)|f (n)Φ(1)x (n)Ψx (n)| < const.

(qv + )ndn

for n ≥ 0.

On the other hand, when n is negative we chose the contour C1 as a circle of radius v − to get a bound like(2)
|f (n)Φ(1)x (n)Ψx (n)| < const.

1(v − )n1 − q n /ζ d n

for n < 0.

In both cases condition (4.16) allows us to select  small enough to guarantee exponential
decay in |n|.
181

C.2

Construction of contours

Here we discuss the construction of the steep descent contour C and that of the steep
ascent contour D which were used in the asymptotic analysis of the stationary higher spinvertex model in section 4.2.
Proposition C.2.1. Consider fixed real numbers
0 < v < ς,

0<q<1

and assume that
ς < inf {ξk sk } ≤ sup{ξk sk } < ∞
k≥2

and

k≥2

0 ≤ s2k < 1,

for all k ≥ 2.

Take also a number ρ < ς and define the contourCρ = {ρeiϑ |ϑ ∈ [0, 2π)}.
Then, for ρ sufficiently close to ς we havedRe{g(ρeiϑ )} < 0dϑ

for 0 < ϑ < π,

(C.5)

where g is given in (4.80).
Remark C.2.2. The result of Proposition C.2.1 implies that Cρ is a steep descent contour
for Re(g) and in particular
1. maxz∈Cρ Re{g(z)} = g(ρ);
2. maxz∈Cρ |z| = ρ.
This easily follows from (C.5) and from the fact that g(z) = g(z), which implies that
Re(g) is symmetric with respect to the real axis.
Proof. Evaluating the derivative we have
!
J−1
i
X
d
q
uρ
Re{g(ρeiϑ )} = sin ϑ κdϑ
1 + q 2i u2 ρ2 − 2q i uρ cos ϑi=0j s2 ρ
jρ
q
q
x
k
1 XX

ξk sk
ξk sk
+ sin ϑ−.
 j 2
 j 2 2
j s2 ρ
jqqsρ
x k=2 j≥0
q ρ
q ρ
k
k
1 + ξk sk − 2 ξk sk cos ϑ 1 + ξk sk− 2 ξk sk cos ϑ(C.6)

Each termq i uρ
,1 + q 2i u2 ρ2 − 2q i uρ cos ϑ182

has a maximum in ϑ = 0 due to the fact that u and ρ have opposite sign, and so does eachsingle one of the summands in the double summation in (C.6), since the generic functiona
1 + a2 − 2a cos ϑ−aσ
1 + a2 σ 2 − 2aσ cos ϑ

is decreasing in 0 < ϑ < π, provided that 0 < a, σ < 1. Now, if ρ is taken sufficiently
close to the critical point ς, in a neighborhood of ϑ = 0, the derivative of Re{g(ρeiϑ )}is negative by construction and, thanks to considerations we just made, it stays negative
along the whole half circle.
The construction of an explicit steepest descent contour D for a general choice ofparameters q, Ξ, S becomes more complicated. Therefore we use the next Proposition
both to exhibit a contour in a rather simple setting and to implicitly deduce conditions
on q, Ξ, S under which our arguments of Section 4.2 are perfectly well posed.
Proposition C.2.3. For each choice of
0 < ς < a,u < 0,

0 < σ < 1,there exist constants Ra , Rσ , Rq > 0, such that for each choice of parameters {ξk }k≥2 , {sk }k≥2 ,q satisfying
ς < inf {ξk sk },k≥2|ξk sk − a| < Ra ,|s2k − σ| < Rσ ,q < Rq ,we are able to construct a complex contour D encircling the set {ξk sk }k≥2 , for which
1. minz∈D Re{g(z)} = g(ς);
2. minz∈D |z| = ς,where g is given in (4.80).
Proof. To show this result we essentially make use of a continuity argument. We start
studying the case whenq = 0,

ξk , sk = as2k = σfor all k ≥ 2.

With this choice of parameters the function g becomesa−z+ O(x−1 ),g(z) = −η log(z) + κ log(1 − uz) + log
a − σz(C.7)

where we can neglect the contribution of the O(x−1 ) term as we are interested in thisresult only in the limiting case of x → ∞. We define the contour D to be the level curve

 



a−z
a−ς
D = z : Re log= log,
a − σza − σς
183

which is a circle and admit the parametrizationς + ρ + ρeiϑ | ϑ ∈ [0, 2π) ,with the radius ρ being

a2 − aς − aςσ + ς 2 σ.
a + aσ − 2ςσWe also report that the leftmost and rightmost extremes of the contour D are respectively
ς and ς + 2ρ and one can easily find that the latter satisfies the inequalityρ=ς + 2ρ ≤

2a
.
1+σ

(C.8)

Along the curve D we are able to calculated
Re g(ς + ρ + ρeiϑ )dϑand to analytically show that its only critical points are θ ∈ Zπ. More specifically,
substituting in (C.7) the correct expressions of coefficients η, κ given in (4.35)
η=

we getaς 2 (1 − σ)(a − a2 u + aσ − 2ςσ + ς 2 uσ),(a − ς)2 (a − ςσ)2a(1 − ςu)2 (1 − σ)(a2 − ς 2 σ)κ=−,u(a − ς)2 (a − ςσ)2

(C.9)
(C.10)

d1
Re g(ς + ρ + ρeiϑ ) = sin ϑ(1 + cos ϑ).
dϑP (cos ϑ)

In the last expression P is a polynomial of degree two in the argument and we see thatzeros are only achieved on the real axis for θ = kπ for k ∈ Z. We can at this point readily
verify that, along D the real part of g assumes a minimum at z = ς and a maximum atz = ς + 2ρ as the function−η log(y) + κ log(1 − uy)is increasing for y > ς, and one can check this by direct inspection of its first derivative,by making use of expressions (4.35) for η and κ.
We can now use the fact that g is continuous in the parameters Ξ, S, q for z belonging
to D and the fact that, by construction, it will always have a critical point in z = ς, tostate the existence of neighborhoods respectively of a, σ and 0 in which every choice ofξk sk , s2k and q will preserve the steepest descent properties 1 and 2.

C.3

Triangular Sums

Here we write down a number of identities of summations of certain symbols ak,` , bα usedin the proof of Proposition 5.3.2. Fix a positive integer N , and assume that the symbols
184

bα , α = 1, . . . , N commute with each other. Let ak,` be
if 0 = k, or ` = N + 1;
0
if 0 ≤ k = ` ≤ N ;ak,` = 1

∈ R
else.

Proposition C.3.1. For any N ≥ 1, the following identities holdX

0≤k<`≤NX

0≤k<`≤NX

0≤k<`≤NX

0≤k<`≤N0≤k<`≤NX

0≤k<`≤Naj,j+1 ;j=1(` − k + 1)(ak+1,`+1 − ak,`+1 ) − (` − k − 1)(ak+1,` − ak,` ) =(ak+1,` − ak,` − ak+1,`+1 + ak,`+1 )

X̀

bα =

N
X

N
−1
X

aj,j+1 ;

j=1

bα ;

α=1

α=k+1

(` − k − 1)2 (ak+1,` − ak,` ) − (` − k + 1)2 (ak+1,`+1 − ak,`+1 )
=−

X

N
−1
X

ak+1,` − ak,` − ak+1,`+1 + ak,`+1 = N −(ak+1,` − ak,` − ak+1,`+1 + ak,`+1 )


N
−1
X
j=1

aj,j+1 − 2X

k+1≤α,β≤lXak,` ;

1≤k<`≤Nbα b β =NX
α=1b2α + 2X

ak,` bk b` ;1≤k<`≤N

 X̀
(` − k + 1)(ak+1,`+1 − ak,`+1 ) − (` − k − 1)(ak+1,` − ak,` )
bα
α=k+1

=

N
X

1≤k<`≤Nak,` (bk − b` ).

Proof. All these identities are elementary and can be proven by induction in a straightforward way.

185

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