Asymptotic expansion of smooth functions in polynomials in deterministic matrices and iid GUE matrices

TL;DR

This paper establishes an asymptotic expansion for the expected trace of smooth functions of polynomials in GUE and deterministic matrices, connecting random matrix behavior with free probability and spectral properties.

Contribution

It provides explicit asymptotic expansions for traces of functions of polynomials in GUE matrices, linking them to free probability and spectral analysis.

Findings

Asymptotic expansion with explicit constants  in terms of free probability

Vanishing coefficients when the function's support and spectrum are disjoint

Eigenvalues concentrate near the spectrum of the free limit

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 non-commutative polynomial, that is for any $N$, $P(X^N)$ 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 $$ \mathbb{E}\left[\frac{1}{N}\text{Tr}\left( f(P(X^N,Z^N)) \right)\right]\ =\ \sum_{i=0}^k \frac{\alpha_i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal{O}(N^{-2k-2}) .$$ 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, for all $i$, $\alpha_i^P(f,Z^N)=0$. 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 from the spectrum of $P(x,Z^N)$.