Asymptotic Expansion of Smooth Functions in Polynomials in Deterministic Matrices and iid GUE Matrices

Abstract

Let \(X^N\) be a family of \(N\times N\) independent GUE random matrices, \(Z^N\) a family of deterministic matrices, P a self-adjoint noncommutative polynomial, that is for any N, \(P(X^N,Z^N)\) is self-adjoint, f a smooth function. We prove that for any k, if f is smooth enough, there exist deterministic constants \(\alpha _i^P(f,Z^N)\) such that $$\begin{aligned} \mathbb {E}\left[ \frac{1}{N}\text {Tr}\left( f(P(X^N,Z^N)) \right) \right] \ =\ \sum _{i=0}^k \frac{\alpha _i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal {O}(N^{-2k-2}) . \end{aligned}$$Besides, the constants \(\alpha _i^P(f,Z^N)\) are built explicitly with the help of free probability. In particular, if x is a free semicircular system, then when the support of f and the spectrum of \(P(x,Z^N)\) are disjoint, \(\alpha _i^P(f,Z^N)=0\) for all \(i\in \mathbb {N}\). As a corollary, we prove that given \(\alpha <1/2\), for N large enough, every eigenvalue of \(P(X^N,Z^N)\) is \(N^{-\alpha }\)-close to the spectrum of \(P(x,Z^N)\).