Random matrices with log-range correlations, and log-Sobolev inequalities
Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 2, pp. 207-232.

Let X N be a symmetric N×N random matrix whose N-scaled entries are uniformly square integrable. We prove that if the entries of X N can be partitioned into independent subsets each of size o(logN), then the empirical eigenvalue distribution of X N , minus its mean, converges weakly to 0 in probability; hence if the averaged empirical eigenvalue distribution converges to a law, the empirical spectral distribution converges to this limit law weakly in probability. If the entries are bounded, the convergence is almost sure; if the entries are Gaussian, we prove almost sure convergence with larger blocks of size o(N 2 /logN). This significantly extends the best previously known results on convergence of eigenvalues for matrices with correlated entries, where the partition subsets are blocks and of size O(1). We also prove the strongest known convergence results for eigenvalues of band matrices.

We prove these results by developing a new log-Sobolev inequality which generalizes the second author’s introduction of mollified log-Sobolev inequalities: we show that if Y is a bounded random vector and Z is a standard normal random vector independent from Y, then the law of Y+t 1/2 Z satisfies a log-Sobolev inequality for all t>0, and we give bounds on the optimal log-Sobolev constant.

Published online:
DOI: 10.5802/ambp.396
Todd Kemp 1; David Zimmermann 

1 Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Todd Kemp; David Zimmermann. Random matrices with log-range correlations, and log-Sobolev inequalities. Annales mathématiques Blaise Pascal, Volume 27 (2020) no. 2, pp. 207-232. doi : 10.5802/ambp.396. https://ambp.centre-mersenne.org/articles/10.5802/ambp.396/

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