Asymptotic expansion of smooth functions in deterministic and iid Haar unitary matrices, and application to tensor products of matrices

TL;DR

This paper develops an asymptotic expansion for the expected trace of smooth functions of polynomials in Haar unitary matrices and deterministic matrices, linking random matrix behavior to free probability and eigenvalue spectrum approximation.

Contribution

It provides explicit asymptotic expansions for traces involving Haar unitaries and deterministic matrices, with applications to eigenvalue localization and norm convergence, using free probability techniques.

Findings

Asymptotic expansion of expected trace with explicit coefficients

Eigenvalues of polynomial in Haar unitaries are close to free counterparts

Convergence of polynomial norms under specific size constraints

Abstract

Let $U^N$ be a family of $N\times N$ independent Haar unitary random matrices and their adjoints, $Z^N$ a family of deterministic matrices, and $P$ a self-adjoint noncommutative polynomial, i.e. for any $N$, $P(U^N,Z^N)$ 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 $$ \mathbb{E}\left[\frac{1}{N}\text{Tr}\left( f(P(U^N,Z^N)) \right)\right]\ =\ \sum_{i=0}^k \frac{\alpha_i^P(f,Z^N)}{N^{2i}}\ +\ \mathcal{O}(N^{-2k-2}) .$$ Besides, the constants $\alpha_i^P(f,Z^N)$ are built explicitly with the help of free probability. As a corollary, we prove that given $\alpha<1/2$, for $N$ large enough, every eigenvalue of $P(U^N,Z^N)$ is $N^{-\alpha}$-close to the spectrum of $P(u,Z^N)$ where $u$ is a $d$-tuple of free Haar unitaries. We also prove the convergence of the norm of any polynomial $P(U^N\otimes I_M, I_N\otimes Y^M)$ as long as the family $Y^M$ converges strongly and that $M\ll N \ln^{-3}(N)$.