Stochastic six vertex model
Ivan Corwin (Columbia University)

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Goals of first hour
Physical goal: Uncover nonequilibrium Kardar-Parisi-Zhang (KPZ)
universality class behavior in the equilibrium six vertex model (6V).
Growth dynamics of cancer cell colonies and their comparison
with noncancerous cells. PRE, 2012, Huergo et. al.

Kardar-Parisi-Zhang class

Square ice in graphene nanocapillaries, Nature 2015, Algara-Siller et. al.

Six vertex model

Mathematical goal: Describe how to analyze the stochastic six vertex
model (S6V) via Markov dualities and Bethe ansatz methods.
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Six vertex model [Pauling '35], [Slater '41], [Lieb '67]
Square-ice model based on six orientationsof H20. Other molecules (e.g. KH2PO4) haveunequal binding energy. Led [Slater '41] to
the general six vertex model.

Probability proportional
to product of weights.

Configurations(two equivalent
ways to draw them)

Weights

What happens in the large system limit? How do weights matter?
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Gibbs states
Infinite volume 'Gibbs states' Given boundary of aand free energies are key
to answer these questions.

subregion, probabilityof inside proportional
to product of weights

Should be limits of the model on a torus.

For (a,b,c) fixed, choosing different H and V external fields should
lead to (possibly) different Gibbs states. The phase diagram of such
Gibbs states is mostly conjectural and relies upon a key parameter.
Stochastic six vertex 1 Page 4

Phase diagrams
Ferroelectric:

From: Lectures on the integrability of
the 6-vertex model, 2010, Reshetikhin.

Disordered:

B1

A2

B1

Antiferroelectric:
A2

B1

A2

C

A1

B2

A1

B2

A1

Ordered Gibbs states:

A1

B1

B2

A2
Stochastic six vertex 1 Page 5CB2

Phase diagrams
Ferromagnetic:

From: Lectures on the integrability of
the 6-vertex model, 2010, Reshetikhin.

Disordered:

B1

A2

Antiferromagnetic:

B1

A2

B1

A2

C

What
happensinthedisorderedphase,
or
at
A
B
A
B
A
B
1

2

1

2

1

2

its boundary?
How
disorderedisdisordered?
Ordered Gibbs states:

A1

B1

B2

A2
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CDisordered phase (free fermion case)
Points in disordered phase lead to Gibbs states with various
average horizontal and vertical line densities.
[Nienhuis '84] conjectured that disordered states
have Gaussian free field height function fluctuations.
[Kenyon '01] proved results for free fermion case (Portion of a
Gibbs state
for the aztec
tiling model
Domain wallboundary conditions

Stochastic six vertex 1 Page 7).

Stochastic pointFor

the 'conical points' in the phase diagram correspondwith a one-parameter family of explicit 'stochastic' Gibbs states.
[Jayaprash-Sam '84] [Bukman-Shore '95] [Aggarwal '16]
B1

Stochastic

A2

Gibbs states
Disordered
Gibbs states

Conical

Disordered

points
A1

Gibbs states

B2

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Stochastic six vertex modelOn first quadrant, for, special choice of vertex weights

yields stochastic six vertex model (S6V) [Gwa-Spohn '92].
Markov update provides interacting particle system interpretation.
Path

Particle

picturepictureStochastic six vertex 1 Page 9

Stochastic Gibbs states
Let

and (

,

) be solutions to

Bernoulli product measure (the y-axis andonon the x-axis)

is stationary [Aggarwal '16]
and hence produces an infinite
volume 'stochastic Gibbs state'.
Theorem [Bukman-Shore '95] [Aggarwal '16]:
Stochastic Gibbs states (above) are conical point
Gibbs states for the symmetric 6V model when.

,
Stochastic six vertex 1 Page 10.

Stationary S6V height fluctuations
Define the height function (zero at the origin):

0

1

2 3

1

2

3 4
3 4 5

Theorem [Aggarwal '16]: For fixed
the stochastic Gibbs state height function
fluctuates like distance1/2 with Gaussian distribution, except along the 'characteristic
direction' where it's like distance1/3 with stationary KPZ distribution [Baik-Rain '01].

characteristic direction comes from the
Hamilton-Jacobi hydrodynamic limit flux
(essentially

Stochastic six vertex 1 Page 11

as a function of).

Stationary S6V height fluctuations

Compare to conjectural Gaussian
free
0
1 2 3
Define the height function (zero at the origin):

1 2with
3 4
field disordered phase behavior
3 4 5Theorem [Aggarwal '16]: For
fixed
the stochastic Gibbs state height function
logarithmic
scale
fluctuations.

fluctuates like distance1/2 with Gaussian distribution, except along the 'characteristic
direction' where it's like distance1/3 with stationary KPZ distribution [Baik-Rain '01].

characteristic direction comes from the
Hamilton-Jacobi hydrodynamic limit flux
(essentially

Stochastic six vertex 1 Page 12

as a function of).

Step initial data S6V height fluctuations
Theorem [Borodin-C-Gorin '14]: For step initial data S6V

where the limit shape is

The fluctuations around the limit shape are given by

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0

1

1

2

2 3

3 4
3 4 5

SPDE limit of S6V
Theorem [C-Ghosal-Shen-Tsai '18]: Let

with

.

The stationary initial data S6V height function converges (after centering andscaling) along the characteristic directions to the

KPZ equation

stationary (Brownian initial data) solution to the
KPZ equation:.

Stochastic Gibbs states converge to stationary
solutions to the stochastic Burgers equation!

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Recap and what's next
• Gibbs states arise from 6V on torus with external fields. Mapping

between field strength and Gibbs state line densities is not simple.
• Disordered states should have GFF and log-correlated fluctuations.
• Stochastic Gibbs states arise at conical point. Fluctuations have
1/3 KPZ exponent in characteristic directions, and the entire
field admits a limit whento the stationary KPZ equation.

• There are other KPZ class / equation convergence results.
• Rest of the talk will focus on two methods (Markov duality and
Bethe ansatz) which play important roles in these type of results.

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Markov duality
Definition: Two Markov processes
if for all

and
and

Theorem [C-Petrov '15]: The S6V particle process

reversed S6V k-particle process
where

are dual with respect to:.

and the independent, space

are dual with respect to

and

,
.

Such dualities can be proved directly (as above or in [Borodin-C-Sasamoto '12]…), inductively ([Lin '19]) or
based on quantum group symmetries ([Schutz '95], [Carinci-Giardina-Redig-Sasamoto '16], [Kuan '17]…)
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Microscopic stochastic heat equation
Definition: The Cole-Hopf solution to the KPZ equationis,wheresolves the stochastic heat equation (SHE).

S6V duality implies that-->.

solves a discrete SHE with an explicit martingale whose

quadratic variation involves the k=2 duality function.
Key challenge in convergence to SHE is to control the martingale.
• [Bertini-Giacomin '95] does this for ASEP via complicated identity (doesn't work for S6V).
• [C-Ghosal-Shen-Tsai '17] uses 2-particle duality and Bethe ansatz.
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(Coordinate) Bethe ansatz[Borodin-C-Gorin '14]: Explicit formulas for transition probabilities
for k-particle S6V (in spirit of [Tracy-Widom '07] and [Lieb '67])

where

and

.

Explicit formulas like these
are also starting points for
KPZ universality asymptotics

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Plancherel theory
Left/right eigenfunctions diagonalize k-particle S6V transition matrix:

For

and

define,and.
Plancherel theory [Borodin-C-Petrov-Sasamoto '14]:
The forward transformare mutual inverses so thatand the backward transform

andStochastic six vertex 1 Page 19.

Some extensions• Symmetric functions:○ [Borodin '14], [Borodin-Petrov '16], [Borodin-Wheeler '17] prove Cauchy identities,
Pieri and branching rules for 'spin Hall-Littlewood' and 'spin q-Whittaker' functions.

• Fusion:○ [C-Petrov '15] (building on [Kulish-Reshetikhin-Sklyanin '81] and [Mangazeev '14])
introduce higher-spin stochastic vertex models (Beta RWRE arises from this).

• Elliptic:○ [Borodin '16] lifts to elliptic level 'dynamic S6V' and 'dynamic ASEP'. [Borodin-C '17]
prove duality for dynamic ASEP and [C-Ghosal-Matetski '19] prove SPDE limit.
[Aggarwal '16] gives higher-spin dynamic models.

• High rank:○ [Kuniba-Mangazeev-Maruyama-Okado '16] introduce high rank models and [Kuan '17]
proves their duality. [Borodin-Wheeler '18] develop their symmetric function theory.
• And much more…
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Summary• Six vertex model has an interesting (mostly conjectural) phase diagram.
• Disordered Gibbs states are expected to have GFF fluctuations.
• Using the stochastic six vertex model, we can construct the oneparameter family of stochastic Gibbs states which arise at theconical point in the phase diagram.
• Stochastic Gibbs states show KPZ universality class fluctuations
along their characteristic direction, and when

they

converge to the stationary solutions to the KPZ equation.
• Duality and Bethe ansatz are key tools in proving both results.

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Stochastic six vertex model
Ivan Corwin (Columbia University)

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Goals of second hour
• Prove a stochastic heat equation (SHE) Laplace transform formula:

whereis the fundamental solution to the SHE:

andis the Airy2 point process.

• Tomorrow: Tsai will use this identity as the parting point to derive
large deviations and tails for the KPZ equation.

• Today: We will derive this result using a combination of two tools:○ Yang-Baxter equation and Macdonald processes
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Airy2 point process• Edge limit of GUE:
Airy2
point
process• Airy2 point process has other characterizations:○ A determinantal point process with a simple explicit kernel.
○ The spectrum of the stochastic Airy operator.
○ The limit of many other point processes, e.g. the Schur measurethat we will encounter later.
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How to prove the identity?

• [Borodin-Gorin '16] proved this as an easy corollary of the SHE
Fredholm determinant formula [Sasamoto-Spohn '10], [CalabreseLe Doussal-Rosso '10], [Dotsenko '10], [Amir-C-Quastel '10].
• Where does such a formula come from? Can compute SHE moments
via 'replica trick' (a version of Markov duality). Moments DO NOT
determine the distribution of the SHE, so that route is not rigorous!
• Proof: We lift to a discrete regularization, the S6V model and use a
non-trivial relationship between S6V and measures on partitions.
There are other approaches (including duality) I will not discuss…
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Inhomogeneous stochastic six vertex model• Consider an inhomogeneous version of S6V with weights

with ax for column x and by is for row ysuch that axby and t in [0,1).
Fix step initial data and definea height function H(x,y) as shown(this picture is flipped versus earlier for convenience)

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RecallA lifting of the identity

Limit asand
The identity!

Limit 1: Convergence result from S6V to KPZ/SHE that we alreadysaw along with limit of the Pochhammer symbol to an exponential.

Identities 2 and 3: We will focus on these. Identity 2 uses ideas from
[Betea-Wheeler-Zinn-Justin 14] + [Borodin-Bufetov-Wheeler 16].
Limit 4: Follows from convergence of explicit determinantal kernels.
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t-Boson vertex model
Consider the following vertex weights [Tsilevich '06] with arbitrary
vertical lines and 0/1 horizontal lines, subject to line conservation:

Here t and a or in [0,1). We can put together weights like thiswhere the internal line configurations are summed over.
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Skew Hall-Littlewood polynomials
Taking, the weight ofis non-zero only when

there is no incoming arrow on the right.
Line conservation implies

Partition notation:

partition lengths satisfy.

Skew Hall-Littlewood polynomial
Multivariable skew Hall-Littlewood
polynomial

involves

stacking rows with variables a1,…,aMand summing weights over all possible

internal configurations and j1,…,jM.
Stochastic six vertex 1 Page 29= weight fromto.

Yang-Baxter equation
The relationship between Hall-Littlewood polynomials and the S6V
model can be seen from the Yang-Baxter equation:

External lines are fixed and internal lines are summed over.
oSpectral variables flip and the X's are S6V vertices rotated by 45 .
Iterating:

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Q polynomials and limiting Yang-Baxter equation-1

YBE involves a and b . We introduce a dual set of t-Boson weights(in salmon) by replacing a by b-1 and multiplying through by b.

Composing and takeonly

non-zero with right incoming line:
If we take thelimit of the YBE, we arrive at:

Weightof

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Hall-Littlewood process and Cauchy identity
Fixing parameters a1,…,aM and b1,…bN, define the Hall-Littlewood
process to be the probability measure on sequences of partitions:

Normalization is given by the
Cauchy-Littlewood identity:

In terms of the t-Boson vertex model,
the Hall-Littlewood process is the weightof this configuration:?'s relate to change in length of partitions.
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Hall-Littlewood process lengths
Consider the probability of seeing lengths T(1),…,T(M+N) under theHall-Littlewood process. This equals the sum of weights of all
configurations like

Now, use the YBE to sequentially swap b and a rows:

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Enter the S6V height function
After swapping the order of all a's and b's, the normalizing constant
has been fully absorbed and we are left equality to the weight of

The weight of the left side of the picture is 1 and the right sideweight is precisely the probability of seeing the given sequence of
output lines for the step initial data S6V model.
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We have proved identity 2
Recall identity 2:• We used the Yang-Baxter equation to match the distribution of theHall-Littlewood process lengths to the distribution of the output
lines for the S6V model. Identity 2 follows readily from this.
• The marginal of the Hall-Littlewood measure on a single
intermediate partition is the Hall-Littlewood measure on a partition

with Hall-Littlewood polynomials P and Q, and same Z as before.
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Identity 3: Enter the Macdonald measure
Replacing P and Q by Macdonald symmetric polynomials depending
on q and t in [0,1), [Borodin-C '11] define the Macdonald measure:

The normalization is now given by the Cauchy-Littlewood identity:

Macdonald polynomials have many degenerations, including○ Hall-Littlewood polynomials when q=0,
○ Schur polynomials when q=t.
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q-independence
Define the Macdonald difference operator which act the a-variables.

Here, u is arbitrary andshifts ai to qai. The eigenrelationwithnot only defines the polynomials, but also enables us to calculate.
It is easy to see that the right side is, in fact, q-independent!
Equating q=0 (Hall-Littlewood) and q=t (Schur) yields identity 3.
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Recap

To prove an identity between SHE and Airy, we found an identity
between two discrete regularizations, S6V and Schur measure.
○ Identity 2 relates S6V to the Hall-Littlewood process using theYang-Baxter equation for t-Bosons.
○ Identity 3 relate the Hall-Littlewood and Schur measure using afurther lifting to Macdonald measures.
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Some extensions• Hall-Littlewood RSK:
○ [Bufetov-Matveev '17] provide a Markov chain on interlacing partitions which preserves
that class of Hall-Littlewood processes and whose marginal on lengths is the S6V model.

• Yang-Baxter fields and bijectivization:○ [Bufetov-Petrov '17] and [Bufetov-Mucciconi-Petrov '19] use Yang-Baxter equation to
construct other Markov chains like above, also for higher-spin models.

• Half-space:○ [Barraquand-Borodin-C-Wheeler '17] provide relation between half-space versions of
the S6V and Hall-Littlewood process (special case of half-space Macdonald process[Barraquand-Borodin-C '18]) and prove half-space identity and asymptotics.

• Gibbsian line ensembles○ [C-Dimitrov '17] interpret S6V and Hall-Littlewood relationship via Gibbsian line
ensembles [C-Hammond '11] and prove predicted KPZ 2/3 transversal exponent.

• And much more…
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Summary of three lectures• What did I do
○ Lecture 1: Conjured and 'solved' the Beta RWRE out of thin air.

○ Lecture 2: 'Solved' S6V via Bethe ansatz diagonalization and Markov duality.
○ Lecture 3: Revealed a key source of solvability, the Yang-Baxter equation, andconnected vertex models to symmetric function measures on partitions.

• What didn't I do?
○ Lots! For example, how does the Beta RWRE arises from S6v? From where does duality
come? How does one do actually perform asymptotics?...

• What will other people do at this program?
○ Imamura: Higher-spin models and another other routes to get Fredholm determinant
formulas for measures in the Macdonald hierarchy.
○ Tsai: How to use the identity we discussed in this lecture to prove KPZ equation tailand large deviation results.
○ Basu: How inputs from integrable probability inform geometric problems in LPP.
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