Boundary Conditions for Scaled Random Matrix Ensembles in the Bulk of the Spectrum

TL;DR

This paper determines boundary conditions for a special tau-function related to the local eigenvalue spacing distribution in large random Hermitian matrices, revealing its specific boundary behavior at a singular point.

Contribution

It identifies the boundary conditions of the tau-function associated with the bulk eigenvalue spacing distribution in large random matrices, especially for the separatrix case.

Findings

Tau-function characterized by boundary conditions at s=0.

Boundary conditions specify the separatrix or truncated type.

Results connect spectral averages to Painlevé V system.

Abstract

A spectral average which generalises the local spacing distribution of the eigenvalues of random $ N\times N $ hermitian matrices in the bulk of their spectrum as $ N\to\infty $ is known to be a $\tau$-function of the fifth Painlev\'e system. This $\tau$-function, $ \tau(s) $, has generic parameters and is transcendental but is characterised by particular boundary conditions about the singular point $s=0$, which we determine here. When the average reduces to the local spacing distribution we find that $\tau$-function is of the separatrix, or partially truncated type.