Coupling of different solvable ensembles of random matrices

TL;DR

This paper explores how to couple different solvable random matrix ensembles, such as complex and unitary matrices, with classical ensembles, providing explicit expressions and integrable structure insights.

Contribution

It introduces methods to couple various solvable random matrix ensembles and demonstrates their integrability via tau functions of classical equations.

Findings

Explicit expressions for coupled ensembles derived

Partition functions identified as tau functions of integrable systems

Examples include complex, unitary, orthogonal, and symplectic ensembles

Abstract

Explicit expressions for multimatrix models with complex and unitary matrices allows to couple these models with well-known unitary, orthogonsl and sympletic ensembles. We consider examples of such mixed ensembles which are solvable in the sense that the partition functions of such ensembles can be considered as tau functions of the classical integrable equations.