Asymptotic expansions for the Gaussian Unitary Ensemble

TL;DR

This paper provides an alternative analytical proof for the asymptotic expansion of the expected trace of functions of GUE matrices, detailing the coefficients as distributions and extending to covariance analysis, especially for resolvent functions.

Contribution

It offers a new proof method for the asymptotic expansion of GUE matrix traces and characterizes the coefficients as distributions, also deriving covariance expansions for resolvent functions.

Findings

Derived explicit asymptotic expansion for E{tr_n(g(X_n))} in GUE

Characterized expansion coefficients as distributions

Extended analysis to covariance of trace functions

Abstract

Let g:{\mathbb R} --> {\mathbb C} be a C^{\infty}-function with all derivatives bounded and let tr_n denote the normalized trace on the n x n matrices. In the paper [EM] Ercolani and McLaughlin established asymptotic expansions of the mean value E{tr_n(g(X_n))} for a rather general class of random matrices X_n,including the Gaussian Unitary Ensemble (GUE). Using an analytical approach, we provide in the present paper an alternative proof of this asymptotic expansion in the GUE case. Specifically we derive for a GUE random matrix X_n that E{tr_n(g(X_n))}= \frac{1}{2\pi}\int_{-2}^2 g(x)\sqrt{4-x^2} dx +\sum_{j=1}^k\frac{\alpha_j(g)}{n^{2j}}+ O(n^{-2k-2}), where k is an arbitrary positive integer. Considered as mappings of g, we determine the coefficients \alpha_j(g), j\in{\mathbb N}, as distributions (in the sense of L. Schwarts). We derive a similar asymptotic expansion for the covariance Cov{Tr_n[f(X_n)],Tr_n[g(X_n)]}, where f is a function of the same kind as g, and Tr_n=n tr_n. Special focus is drawn to the case where g(x)=1/(z-x) and f(x)=1/(w-x) for non-real complex numbers z and w. In this case the mean and covariance considered above correspond to, respectively, the one- and two-dimensional Cauchy (or Stieltjes) transform of the GUE(n,1/n).