All the lowest order PDE for spectral gaps of Gaussian matrices

TL;DR

This paper derives all the lowest order PDEs governing spectral gap probabilities for Gaussian Unitary Ensembles using a unified approach based on Tracy-Widom equations, simplifying previous methods and providing explicit equations.

Contribution

It introduces a general form of Tracy-Widom equations that enables derivation of all lowest order PDEs for spectral gaps without higher-order intermediates, specifically applied to GUE.

Findings

Explicit third order PDEs for GUE gap probabilities.

A second order PDE for GUE with multiple spectral endpoints.

Unified derivation method simplifies obtaining PDEs for spectral gaps.

Abstract

Tracy-Widom (TW) equations for one-matrix unitary ensembles (UE) (equivalent to a particular case of Schlesinger equations for isomonodromic deformations) are rewritten in a general form which allows one to derive all the lowest order equations (PDE) for spectral gap probabilities of UE without intermediate higher-order PDE. This is demonstrated on the example of Gaussian ensemble (GUE) for which all the third order PDE for gap probabilities are obtained explicitly. Moreover, there is a {\it second order} PDE for GUE probabilities in the case of more than one spectral endpoint. This approach allows to derive all PDE at once where possible, while in the method based on Hirota bilinear identities and Virasoro constraints starting with different bilinear identities leads to different subsets of the full set of equations.