Hermitian, symmetric and symplectic random ensembles: PDEs for the distribution of the spectrum

TL;DR

This paper derives nonlinear PDEs governing the spectral distribution probabilities for Hermitian, symmetric, and symplectic random matrix ensembles, connecting them to integrable systems like Toda, KP, and Pfaff-KP equations.

Contribution

It introduces a unified method to obtain PDEs for spectral distributions across classical ensembles using integrable lattice equations and Virasoro constraints.

Findings

Derived PDEs for Gaussian, Laguerre, Jacobi ensembles

Connected PDEs to integrable systems like Toda and KP

Extended results to symmetric and symplectic ensembles

Abstract

Given the Hermitian, symmetric and symplectic ensembles, it is shown that the probability that the spectrum belongs to one or several intervals satisfies a nonlinear PDE. This is done for the three classical ensembles: Gaussian, Laguerre and Jacobi. For the Hermitian ensemble, the PDE (in the boundary points of the intervals) is related to the Toda lattice and the KP equation, whereas for the symmetric and symplectic ensembles the PDE is an inductive equation, related to the so-called Pfaff-KP equation and the Pfaff lattice. The method consists of inserting time-variables in the integral and showing that this integral satisfies integrable lattice equations and Virasoro constraints.