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 $\sqrt{N}$-scaled entries are uniformly square integrable. We prove that if the entries of ${X}_{N}$ can be partitioned into independent subsets each of size $o\left(logN\right)$, 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 $o\left({N}^{2}/logN\right)$. 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 $O\left(1\right)$. 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 $\mathbf{Y}$ is a bounded random vector and $\mathbf{Z}$ is a standard normal random vector independent from $\mathbf{Y}$, then the law of $\mathbf{Y}+{t}^{1/2}\mathbf{Z}$ 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
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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