Various equations for gap probabilities of coupled Gaussian matrices

TL;DR

This paper derives and compares various PDEs satisfied by joint eigenvalue spacing probabilities of coupled Gaussian matrices, revealing new equations and simplifying existing ones, with connections to Painlevé equations.

Contribution

It introduces new PDEs for coupled Gaussian matrices' eigenvalue probabilities, expanding the understanding beyond the Adler-van Moerbeke equation and simplifying the equations.

Findings

Derived all third-order PDEs for largest eigenvalues of coupled GUE.

Compared Tracy-Widom and Adler-Shiota-van Moerbeke approaches.

Identified new PDEs resembling coupled Painlevé IV equations.

Abstract

Versions of Tracy-Widom (TW) and Adler-Shiota-van Moerbeke (ASvM) approaches are applied to derive various partial differential equations (PDE) satisfied by joint eigenvalue spacing probabilities of two coupled Gaussian Hermitian matrices (coupled GUE). All the lowest (third) order PDE satisfied by the probabilities for the largest eigenvalues of two coupled GUE are found, and the results of both approaches are compared. The TW approach allows to derive all PDE at once, while in the ASvM one starting with different bilinear identities leads to different subsets of the full set of equations. An interesting result is that the joint probability for the largest eigenvalues of coupled Gaussian matrices satisfies a number of different PDE, and the previously known Adler-van Moerbeke equation (AvM) [3] is only one of them. Some of the new equations look like "coupled Painlev\'e IV" and have usual Painlev\'e IV equation as one-matrix limit, i.e. when the spectral endpoint of one of the matrices goes to infinity. This is in contrast to the AvM equation, which becomes trivial in this limit. Moreover, the new PDE, which stem from the matrix kernel approach of [23], do not contain derivatives w.r.t. the strength of coupling, unlike the AvM equation. In other words, they contain fewer independent variables and in this sense are simpler.