On an average over the Gaussian Unitary Ensemble

TL;DR

This paper analyzes the asymptotic behavior of a specialized partition function related to the Gaussian Unitary Ensemble, revealing new eigenvalue statistics and their generating functions in the large matrix limit.

Contribution

It computes the leading order of the partition function and its Taylor coefficients for large matrices, introducing new eigenvalue statistics in GUE.

Findings

Asymptotic formula for the partition function in the specified range

Identification of new eigenvalue statistics in GUE

Connection to moment generating functions of singular linear statistics

Abstract

We study the asymptotic limit for large matrix dimension N of the partition function of the unitary ensemble with weight exp(-z^2/2x^2 + t/x - x^2/2). We compute the leading order term of the partition function and of the coefficients of its Taylor expansion. Our results are valid in the range N^(-1/2) < z < N^(1/4). Such partition function contains all the information on a new statistics of the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) that was introduced by Berry and Shukla (J. Phys. A: Math. Theor., Vol. 41 (2008), 385202, arXiv:0807.3474). It can also be interpreted as the moment generating function of a singular linear statistics.