Macdonald processes, quantum integrable systems and the Kardar-Parisi-Zhang universality class

TL;DR

This paper explores the role of integrable probability in understanding the KPZ universality class, highlighting connections with symmetric functions and quantum integrable systems to derive exact formulas and asymptotic behaviors.

Contribution

It elucidates how integrability in probabilistic models related to KPZ arises from links with symmetric functions and quantum integrable systems, enabling precise analysis.

Findings

Exact formulas for observables in KPZ-related models

Asymptotic analysis of large-scale behavior

Connections between integrable probability and quantum systems

Abstract

Integrable probability has emerged as an active area of research at the interface of probability/mathematical physics/statistical mechanics on the one hand, and representation theory/integrable systems on the other. Informally, integrable probabilistic systems have two properties: 1) It is possible to write down concise and exact formulas for expectations of a variety of interesting observables (or functions) of the system. 2) Asymptotics of the system and associated exact formulas provide access to exact descriptions of the properties and statistics of large universality classes and universal scaling limits for disordered systems. We focus here on examples of integrable probabilistic systems related to the Kardar-Parisi-Zhang (KPZ) universality class and explain how their integrability stems from connections with symmetric function theory and quantum integrable systems.