Thermalisation for 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.