On the Variance of the Index for the Gaussian Unitary Ensemble

TL;DR

This paper derives explicit formulas and recurrences for the variance of the index in the Gaussian Unitary Ensemble, linking it to Painlevé equations and hypergeometric functions.

Contribution

It introduces new recurrence relations and exact formulas for the variance of the index, connecting random matrix theory with Painlevé equations.

Findings

Derived simple linear recurrences for the variance.

Obtained integral representations and hypergeometric evaluations.

Linked the index distribution to Painlevé IV tau-functions.

Abstract

We derive simple linear, inhomogeneous recurrences for the variance of the index by utilising the fact that the generating function for the distribution of the number of positive eigenvalues of a Gaussian unitary ensemble is a $\tau$-function of the fourth Painlev\'e equation. From this we deduce a simple summation formula, several integral representations and finally an exact hypergeometric function evaluation for the variance.