Some algebraic structures in KPZ universality

TL;DR

This paper explores algebraic and combinatorial structures like RSK correspondence and representation theory that underpin models in the KPZ universality class, connecting them to stochastic dynamics and random matrix theory.

Contribution

It reviews how combinatorial and algebraic tools are used to analyze solvable KPZ models and develop integrable stochastic processes.

Findings

Connection between combinatorial structures and KPZ models

Construction of integrable stochastic dynamics using representation theory

Link between stochastic processes and random matrix distributions

Abstract

We review some algebraic and combinatorial structures that underlie models in the KPZ universality class.Emphasis is placed on the Robinson-Schensted-Knuth correspondence and its geometric lifting due to A.N.Kirillov. We present how these combinatorial constructions are used to analyse the structure of solvable models in the KPZ class and lead to computation of their statistics via connecting to representation theoretic objects such as Schur, Macdonald and Whittaker functions, Young tableaux and Gelfand-Tsetlin patterns. We also present how fundamental representation theoretic concepts, such as the Cauchy identity, the Pieri rule and the branching rule, can be used, alongside RSK correspondences, and can be combined with probabilistic ideas, in order to construct integrable stochastic dynamics on two dimensional arrays of Gelfand-Tsetlin type, in ways that couple different one dimensional stochastic processes. For example, interacting particle systems, on the one hand, and processes related to eigenvalues of random matrices, on the other, thus illuminating the emergence of random matrix distributions in interacting stochastic processes. The goal of the notes is to expose some of the overarching principles, which have driven a significant number of developments in the field.