From interacting particle systems to random matrices

TL;DR

This paper explores the connection between stochastic growth models in the KPZ universality class and random matrix theory, highlighting how initial conditions influence large-time behavior and the partial correspondence with random matrices.

Contribution

It clarifies the dependence of large-time surface statistics on initial conditions and discusses the partial analogy between growth models and random matrices.

Findings

Large time distribution depends on initial conditions.

Some fluctuation laws originate from random matrix models.

Limit processes relate to dynamical hermitian random matrices.

Abstract

In this contribution we consider stochastic growth models in the Kardar-Parisi-Zhang universality class in 1+1 dimension. We discuss the large time distribution and processes and their dependence on the class on initial condition. This means that the scaling exponents do not uniquely determine the large time surface statistics, but one has to further divide into subclasses. Some of the fluctuation laws were first discovered in random matrix models. Moreover, the limit process for curved limit shape turned out to show up in a dynamical version of hermitian random matrices, but this analogy does not extend to the case of symmetric matrices. Therefore the connections between growth models and random matrices is only partial.