Asymptotic expansions for the Gaussian Unitary Ensemble

Understand

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.