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Title: Almost sure convergence in quantum spin glasses

Abstract

Recently, Keating, Linden, and Wells [Markov Processes Relat. Fields 21(3), 537-555 (2015)] 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 Keating, Linden, and Wells 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 Erdős and Schröder [Math. Phys., Anal. Geom. 17(3-4), 441–464 (2014)].

Authors:
;  [1]
  1. Department of Mathematics, Case Western Reserve University, 10900 Euclid Ave., Cleveland, Ohio 44106 (United States)
Publication Date:
OSTI Identifier:
22479573
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 56; Journal Issue: 12; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; COUPLING; DENSITY OF STATES; ENTROPY; MARKOV PROCESS; RANDOMNESS; SPIN GLASS STATE

Citation Formats

Buzinski, David, and Meckes, Elizabeth. Almost sure convergence in quantum spin glasses. United States: N. p., 2015. Web. doi:10.1063/1.4936956.
Buzinski, David, & Meckes, Elizabeth. Almost sure convergence in quantum spin glasses. United States. https://doi.org/10.1063/1.4936956
Buzinski, David, and Meckes, Elizabeth. 2015. "Almost sure convergence in quantum spin glasses". United States. https://doi.org/10.1063/1.4936956.
@article{osti_22479573,
title = {Almost sure convergence in quantum spin glasses},
author = {Buzinski, David and Meckes, Elizabeth},
abstractNote = {Recently, Keating, Linden, and Wells [Markov Processes Relat. Fields 21(3), 537-555 (2015)] 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 Keating, Linden, and Wells 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 Erdős and Schröder [Math. Phys., Anal. Geom. 17(3-4), 441–464 (2014)].},
doi = {10.1063/1.4936956},
url = {https://www.osti.gov/biblio/22479573}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 12,
volume = 56,
place = {United States},
year = {Tue Dec 15 00:00:00 EST 2015},
month = {Tue Dec 15 00:00:00 EST 2015}
}