Asymptotic freeness through unitaries generated by polynomials of Wigner matrices

TL;DR

This paper investigates how products of functions of polynomials in Wigner matrices and deterministic matrices behave asymptotically, showing that quantum evolution induces asymptotic freeness over time with minimal smoothness assumptions.

Contribution

It introduces a method to approximate and analyze products of functions of polynomials in Wigner matrices, extending asymptotic freeness results to less smooth functions and long-time quantum evolutions.

Findings

Deterministic approximations of products of functions evaluated at polynomials in Wigner matrices.

Control of fluctuations with optimized error terms as matrix size increases.

Asymptotic freeness emerges from quantum evolution of polynomials in Wigner matrices over time.

Abstract

We study products of functions evaluated at self-adjoint polynomials in deterministic matrices and independent Wigner matrices; we compute the deterministic approximations of such products and control the fluctuations. We focus on minimizing the assumption of smoothness on those functions while optimizing the error term with respect to $N$, the size of the matrices. As an application, we build on the idea that the long-time Heisenberg evolution associated to Wigner matrices generates asymptotic freeness as first shown in $[9]$. More precisely given $P$ a self-adjoint non-commutative polynomial and $Y^N$ a $d$-tuple of independent Wigner matrices, we prove that the quantum evolution associated to the operator $P(Y^N)$ yields asymptotic freeness for large times.