Thermalisation for Wigner matrices

Summary

We extend key results from free probability on polynomials to arbitrary Sobolev functions. Applied to time Heisenberg evolution operators this proves the thermalization effect of the unitary group generated by 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.

Paper

2102.09975.pdf