Almost sure convergence in quantum spin glasses

TL;DR

This paper proves that the empirical spectral measure of certain quantum spin glass models converges almost surely to a Gaussian distribution, extending previous ensemble-based results to individual realizations.

Contribution

It establishes almost sure Gaussian convergence of the spectral measure for quantum spin glasses, using concentration and entropy methods, beyond ensemble averages.

Findings

Empirical spectral measures are almost surely Gaussian for large quantum spin glasses.

Extension of Gaussian convergence to spherical models and general coupling geometries.

Uses concentration of measure and entropy techniques for the proof.

Abstract

Recently, Keating, Linden, and Wells \cite{KLW} showed that the density of states measure of a nearest-neighbor quantum spin glass model is approximately Gaussian when the number of particles is large. The density of states measure is the ensemble average of the empirical spectral measure of a random matrix; in this paper, we use concentration of measure and entropy techniques together with the result of \cite{KLW} to show that in fact, the empirical spectral measure of such a random matrix is almost surely approximately Gaussian itself, with no ensemble averaging. We also extend this result to a spherical quantum spin glass model and to the more general coupling geometries investigated by Erd\H{o}s and Schr\"oder.