Thermalisation for Wigner matrices

TL;DR

This paper extends the understanding of Wigner matrices by providing a deterministic approximation for Sobolev functions of large matrices, establishing optimal fluctuation bounds, and demonstrating thermalisation effects in quantum dynamics.

Contribution

It generalizes Voiculescu's theorem from polynomials to Sobolev functions and from traces to individual matrix elements, with applications to quantum thermalisation.

Findings

Optimal error bounds on fluctuations of Sobolev functions of Wigner matrices

Extension of free probability results to more general functions

Demonstration of thermalisation in quantum dynamics with Wigner matrices

Abstract

We compute the deterministic approximation of products of Sobolev functions of large Wigner matrices $W$ and provide an optimal error bound on their fluctuation with very high probability. This generalizes Voiculescu's seminal theorem [Voiculescu 1991] from polynomials to general Sobolev functions, as well as from tracial quantities to individual matrix elements. Applying the result to $\exp(\mathrm{i} tW)$ for large $t$, we obtain a precise decay rate for the overlaps of several deterministic matrices with temporally well separated Heisenberg time evolutions; thus we demonstrate the thermalisation effect of the unitary group generated by Wigner matrices.