R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

Shape Fluctuations and Random Matrices

Quantum inverse scattering method and correlation functions

Consider the process D

 k
 , k = 1,2,…, given by
 
B

 i
 being independent standard Brownian motions. This process describes the limiting behavior “near the edge” in queues in series, totally asymmetric exclusion processes or oriented percolation. The problem of finding the distribution of D. was posed in [GW]. The main result of this paper is that the process D. has the law of the process of the largest eigenvalues of the main minors of an infinite random matrix drawn from Gaussian Unitary Ensemble.

GUEs and queues

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time $\tau$) according to the Tracy-Widom GOE distribution on the $\tau^{1/3}$ scale. This is the first example of KPZ asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. 
Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall-Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to ASEP) using a Yang-Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogs via a refined Littlewood identity.

Stochastic six-vertex model in a half-quadrant and half-line open ASEP

This work is dedicated to $\mathfrak{sl}_{n+1}$-related integrable stochastic vertex models; we call such models coloured. We prove several results about these models, which include the following: 
(1) We construct the basis of (rational) eigenfunctions of the coloured transfer-matrices as partition functions of our lattice models with certain boundary conditions. Similarly, we construct a dual basis and prove the corresponding orthogonality relations and Plancherel formulae; (2) We derive a variety of combinatorial properties of those eigenfunctions, such as branching rules, exchange relations under Hecke divided-difference operators, (skew) Cauchy identities of different types, and monomial expansions; (3) We show that our eigenfunctions are certain (non-obvious) reductions of the nested Bethe Ansatz eigenfunctions; (4) For models in a quadrant with domain-wall (or half-Bernoulli) boundary conditions, we prove a matching relation that identifies the distribution of the coloured height function at a point with the distribution of the height function along a line in an associated colour-blind ($\mathfrak{sl}_2$-related) stochastic vertex model. Thanks to a variety of known results about asymptotics of height functions of the colour-blind models, this implies a similar variety of limit theorems for the coloured height function of our models; (5) We demonstrate how the coloured-uncoloured match degenerates to the coloured (or multi-species) versions of the ASEP, $q$-PushTASEP, and the $q$-boson model; (6) We show how our eigenfunctions relate to non-symmetric Cherednik-Macdonald theory, and we make use of this connection to prove a probabilistic matching result by applying Cherednik-Dunkl operators to the corresponding non-symmetric Cauchy identity.

Coloured stochastic vertex models and their spectral theory

We derive a matrix product formula for symmetric Macdonald polynomials. Our results are obtained by constructing polynomial solutions of deformed Knizhnik-Zamolodchikov equations, which arise by considering representations of the Zamolodchikov-Faddeev and Yang-Baxter algebras in terms of t-deformed bosonic operators. These solutions are generalised probabilities for particle configurations of the multi-species asymmetric exclusion process, and form a basis of the ring of polynomials in n variables whose elements are indexed by compositions. For weakly increasing compositions (anti-dominant weights), these basis elements coincide with non-symmetric Macdonald polynomials. Our formulas imply a natural combinatorial interpretation in terms of solvable lattice models. They also imply that normalisations of stationary states of multi-species exclusion processes are obtained as Macdonald polynomials at q = 1.

https://iopscience.iop.org/article/10.1088/1751-8113/48/38/384001/pdf

Matrix product formula for Macdonald polynomials

We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary condition. We show that, when starting devoid of particles and for a certain boundary condition, the height function at the origin fluctuates asymptotically (in large time τ) according to the Tracy–Widom Gaussian orthogonal ensemble distribution on the τ1/3-scale. This is the first example of Kardar–Parisi–Zhang asymptotics for a half-space system outside the class of free-fermionic/determinantal/Pfaffian models. Our main tool in this analysis is a new class of probability measures on Young diagrams that we call half-space Macdonald processes, as well as two surprising relations. The first relates a special (Hall–Littlewood) case of these measures to the half-space stochastic six-vertex model (which further limits to the ASEP) using a Yang–Baxter graphical argument. The second relates certain averages under these measures to their half-space (or Pfaffian) Schur process analogues via a refined Littlewood identity.

Michael Wheeler

Papers

Stochastic six-vertex model in a half-quadrant and half-line open ASEP

Coloured stochastic vertex models and their spectral theory

Matrix product formula for Macdonald polynomials

Stochastic six-vertex model in a half-quadrant and half-line open asymmetric simple exclusion process

Refined Cauchy/Littlewood identities and six-vertex model partition functions: III. Deformed bosons ☆