A pedestrian's view on interacting particle systems, KPZ universality, and random matrices

TL;DR

This paper explores the connections between interacting particle systems, the KPZ universality class, and random matrix theory, highlighting recent advances and key results in understanding surface growth fluctuations.

Contribution

It provides a comprehensive overview of the KPZ universality conjecture and details Johansson's seminal work linking TASEP current fluctuations to Tracy-Widom distribution.

Findings

Derivation of stationary measure for the simple exclusion process

Explanation of the KPZ equation and its universality

Connection between TASEP fluctuations and Tracy-Widom distribution

Abstract

These notes are based on lectures delivered by the authors at a Langeoog seminar of SFB/TR12 "Symmetries and universality in mesoscopic systems" to a mixed audience of mathematicians and theoretical physicists. After a brief outline of the basic physical concepts of equilibrium and nonequilibrium states, the one-dimensional simple exclusion process is introduced as a paradigmatic nonequilibrium interacting particle system. The stationary measure on the ring is derived and the idea of the hydrodynamic limit is sketched. We then introduce the phenomenological Kardar-Parisi-Zhang (KPZ) equation and explain the associated universality conjecture for surface fluctuations in growth models. This is followed by a detailed exposition of a seminal paper of Johansson that relates the current fluctuations of the totally asymmetric simple exclusion process (TASEP) to the Tracy-Widom distribution of random matrix theory. The implications of this result are discussed within the framework of the KPZ conjecture.