Recursive characterisations of random matrix ensembles and associated combinatorial objects

TL;DR

This paper reviews recursive methods for analyzing classical and product random matrix ensembles, deriving differential equations, recursions, and combinatorial interpretations to understand eigenvalue statistics and correlations.

Contribution

It introduces new recursive schemes and differential equations for classical and product ensembles, linking them to combinatorial ribbon graph interpretations.

Findings

Derived linear differential equations for eigenvalue densities.

Established 1-point recursions analogous to Harer-Zagier.

Connected matrix ensemble statistics to ribbon graph combinatorics.

Abstract

We give an overview of the recursive characterisations of random matrix ensembles that are currently at the forefront of random matrix theory by way of studying two classes of ensembles using two different types of recursive schemes: Established theory on Selberg correlation integrals is used to derive linear differential equations on the eigenvalue densities and resolvents of the classical matrix ensembles, which lead to $1$-point recursions, understood to be analogues of the Harer-Zagier recursion, for the expansion coefficients of the associated $1$-point cumulants, while loop equation analysis is used to recursively compute some leading order correlator expansion coefficients pertaining to certain products of random matrices that have recently come into interest due to their connections to Muttalib-Borodin ensembles and integrals of Harish-Chandra-Itzykson-Zuber type. We also show how the aforementioned differential equations can be used to characterise the large $N$ limiting statistics of the classical matrix ensembles' eigenvalue densities in the global and edge scaling regimes and moreover give a comprehensive review of how the Isserlis-Wick theorem implies ribbon graph interpretations for mixed moments and cumulants of some of the ensembles at hand. It is expected that this thesis will serve as a valuable resource for readers wanting a well-rounded introduction to classical matrix ensembles, matrix product ensembles, $1$-point recursions, loop equations, Selberg correlation integrals, and ribbon graphs.