Recently, Keating, Linden, and Wells 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 KLW to show that in fact, the empirical spectral measure of such a random matrix is almost surely

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... pproximately 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 Erdős and Schröder.