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arXiv · 2102.09975

Thermalisation for Wigner matrices

Giorgio Cipolloni·László Erdős·Dominik Schröder
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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.

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BibTeXRIS

Giorgio Cipolloni, László Erdős, Dominik Schröder. 2021-02-19. Thermalisation for Wigner matrices. https://doi.org/10.1016/j.jfa.2022.109394

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