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

Category(English)

Doctoral Thesis

種別(和文)

論文要旨

Type(English)

Summary

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Thesis summary

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 wecall spin q-Whittaker (sqW) process. The importance of this new model lays in the fact that it
simultaneously generalizes all solvable uncolored models in the KPZ class that were introduced andstudied in the last 20 years. And it does so by embedding the one-dimensional dynamics in a moregeneral 2-dimensional anisotropic growth process weighted by special symmetric polynomials. Thissolves a question that was asked several times in literature and that is to inscribe integrable stochastic
models in more general dynamics on interlacing arrays in the spirit of the Macdonald Processes ofBorodin and Corwin.
Properties of the sqW processes are described by a family of special functions we study extensively:
the sqW polynomials. The sqW polynomials F are a family of symmetric functions introduced in 2017by Borodin and Wheeler. Their basic properties were studied from a combinatorial standpoint leveraging Yang-Baxter relations, but no result about their connection to probabilistic integrable systems orabout their relevant algebraic properties was given. Part of the results presented in this thesis represent
solutions to these issues. We first introduce an improved version of Borodin and Wheeler’s polynomials
and we prove for them symmetry properties and Cauchy summation identities. Most remarkably ourversion of the functions F satisfy eigenrelations with respect to difference operators we identify. Theseresults can be seen as partial generalizations of eigenrelations enjoyed by the Macdonald Polynomials.
We also define an integral scalar product under which these operators are self-adjoint and conjecture
general orthogonality relations for the polynomials F.
Cauchy identities and branching rules of the functions F allow us to define natural probability
measures on sequences of interlacing arrays {λ(i, j) : 1 ≤ i ≤ j}. These are the sqW processes andthey represent generalizations of the Macdonald Processes. To understand the physical meaning ofthese probability measures we implement a sampling algorithm based on couplings of the Yang-Baxterequation. Explicit solutions of the the Yang-Baxter equation, along with explicit realization of suchcouplings allow us to identify the precise dynamics followed by marginals of the sqW processes. We willconsider two specialization of R-matrices solving the Yang-Baxter equation: the spin-Hall Littlewood(sHL) and the sqW specialization. Pairing together sqW and sHL R-matrices we obtain three different
realizations of Yang-Baxter equation (namely sHL/sHL, sqW/sHL and sqW/sqW), resulting in threedifferent classes of dynamics on interlacing arrays. We find that in all cases the rightmost coordinatesof the array {λ(1, j) : 1 ≤ j} evolve as an autonomous one dimensional Markov process. Considering
the dynamics generated by sHL/sHL type Yang-Baxter relations we prove that λ(1, j) is equivalent as
a process to the height function of the stochastic six-vertex model introduced by Gwa and Spohn in1992. For different realization of the Yang-Baxter equation the same type of matching keeps holding.
For sqW/sHL dynamics we match λ(1, j) with the Higher Spin Vertex Model of Corwin and Petrov(2015), while in the sqW/sqW case λ(1, j) becomes equivalent to the q-Hahn push-TASEP introduced
by Corwin-Matveev-Petrov in 2018. We find that the leftmost coordinates of the array {λ(j, j) : 1 ≤ j}too evolve as an autonomous Markov processes in the cases of sqW/sHL and sqW/sqW dynamics. For
the first choice the distribution λ(j, j) is matched again with that of the height function of the HigherSpin Vertex Model. In the remaining case, the the λ(j, j) coordinate of a sqW/sqW process becomes
equivalent to the q-Hahn TASEP introduced by Povolotski in 2013.
We also consider the q → 1 limit of our constructions and this leads to the definition of a new class of
symmetric functions we call spin Whittaker functions. For them we derive integral representations and1

identities generalizing the classical Bump-Stade identities for Whittaker functions. Random processes
weighted by spin Whittaker functions possess as marginals systems of Beta polymers.
We also 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. For this result we perform asymptotic analysis of a Fredholm determinant representation for theq-Laplace transform of the height function. Such representation was derived using elliptic determinant
techniques in a previous work with T.Imamura and T.Sasamoto.

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