A convergent $\frac{1}{N}$ expansion for GUE

TL;DR

This paper proves the convergence of the $1/N$ expansion for GUE linear statistics with entire test functions, enabling exact eigenvalue density recovery and generalizing kernel transform formulas.

Contribution

It establishes the convergence of the $1/N$ expansion for GUE linear statistics with entire functions, providing a resummation method for eigenvalue densities.

Findings

The $1/N$ expansion for GUE averages converges for entire functions of order two and finite type.

The eigenvalue density can be recovered exactly from the expansion coefficients.

A generalized bilateral Laplace transform of the GUE kernel is derived.

Abstract

We show that the asymptotic $1/N$ expansion for the averages of linear statistics of the GUE is convergent when the test function is an entire function of order two and finite type. This allows to fully recover the mean eigenvalue density function for finite $N$ from the coefficients of the expansion thus providing a resummation procedure. As an intermediate result we compute the bilateral Laplace transform of the GUE reproducing kernel in the half-sum variable, generalizing a formula of Haagerup and Thorbj{\o}rnsen.