One-dimensional stochastic growth and Gaussian ensembles of random matrices

TL;DR

This review explores the connection between one-dimensional stochastic growth models, specifically the PNG model, and Gaussian ensembles of random matrices, highlighting their shared fluctuation behaviors and related combinatorial models.

Contribution

It elucidates the relationship between the PNG growth model and Gaussian random matrix ensembles through point processes and multilayer extensions, unifying various models in statistical physics and combinatorics.

Findings

Surface fluctuation distributions match Gaussian ensemble limit laws.

Multilayer PNG and multi-matrix models extend the connection.

Various combinatorial and physical models are equivalent to PNG.

Abstract

In this review paper we consider the polynuclear growth (PNG) model in one spatial dimension and its relation to random matrix ensembles. For curved and flat growth the scaling functions of the surface fluctuations coincide with limit distribution functions coming from certain Gaussian ensembles of random matrices. This connection can be explained via point processes associated to the PNG model and the random matrices ensemble by an extension to the multilayer PNG and multi-matrix models, respectively. We also explain other models which are equivalent to the PNG model: directed polymers, the longest increasing subsequence problem, Young tableaux, a directed percolation model, kink-antikink gas, and Hammersley process.