Application of the $\tau$-Function Theory of Painlev\'e Equations to Random Matrices: PIV, PII and the GUE

TL;DR

This paper extends Tracy and Widom's results on GUE eigenvalue distributions by applying the $ au$-function theory of Painlevé equations PIV and PII, providing new formulas for eigenvalue-related probabilities and densities.

Contribution

It generalizes existing results to evaluate eigenvalue distribution functions using the $ au$-function approach of Painlevé equations, applicable to various classical ensembles.

Findings

Derived new formulas for eigenvalue distribution functions using Painlevé $ au$-functions.

Connected eigenvalue statistics with PIV and PII transcendents.

Applicable framework for other classical matrix ensembles.

Abstract

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\tilde{E}_N(\lambda;a) := \Big < \prod_{l=1}^N \chi_{(-\infty, \lambda]}^{(l)} (\lambda - \lambda_l)^a \Big>$, where $ \chi_{(-\infty, \lambda]}^{(l)} = 1$ for $\lambda_l \in (-\infty, \lambda]$ and $ \chi_{(-\infty, \lambda]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(\lambda;a) := \Big < \prod_{l=1}^N (\lambda - \lambda_l)^a \Big >$. Of particular interest are $\tilde{E}_N(\lambda;2)$ and $F_N(\lambda;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\tau$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\tilde{E}_N(\lambda;a)$ and $F_N(\lambda;a)$ for the other classical matrix ensembles.