On the partial connection between random matrices and interacting particle systems

TL;DR

This paper explores the partial connections between random matrix eigenvalue distributions and interacting particle systems, revealing new links in specific scaling limits and extending known relations to more complex matrix models.

Contribution

It demonstrates that certain eigenvalue correlation functions in matrix minors match those of interacting particle systems under specific conditions, extending the known connections.

Findings

Eigenvalue correlation functions match in specific limits

Connection extends to complex sample covariance matrices

Partial links are fragile and limited to certain distributions

Abstract

In the last decade there has been increasing interest in the fields of random matrices, interacting particle systems, stochastic growth models, and the connections between these areas. For instance, several objects appearing in the limit of large matrices arise also in the long time limit for interacting particles and growth models. Examples of these are the famous Tracy-Widom distribution functions and the Airy_2 process. The link is however sometimes fragile. For example, the connection between the eigenvalues in the Gaussian Orthogonal Ensembles (GOE) and growth on a flat substrate is restricted to one-point distribution, and the connection breaks down if we consider the joint distributions. In this paper we first discuss known relations between random matrices and the asymmetric exclusion process (and a 2+1 dimensional extension). Then, we show that the correlation functions of the eigenvalues of the matrix minors for beta=2 Dyson's Brownian motion have, when restricted to increasing times and decreasing matrix dimensions, the same correlation kernel as in the 2+1 dimensional interacting particle system under diffusion scaling limit. Finally, we analyze the analogous question for a diffusion on (complex) sample covariance matrices.