Random matrices with log-range correlations, and log-Sobolev inequalities

Todd Kemp; David Zimmermann

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

Annales mathématiques Blaise Pascal, Tome 27 (2020) no. 2, pp. 207-232.

Résumé

Let $X_{N}$ be a symmetric $N \times N$ 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 $o (log N)$ , 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 $o (N^{2} / log N)$ . 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.

Publié le : 2021-03-09

DOI : 10.5802/ambp.396

Affiliations des auteurs :

Todd Kemp ¹ ; David Zimmermann

¹ Department of Mathematics University of California, San Diego La Jolla, CA 92093-0112 USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Random matrices with log-range correlations, and {log-Sobolev} inequalities},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {207--232},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
     volume = {27},
     number = {2},
     year = {2020},
     doi = {10.5802/ambp.396},
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Todd Kemp; David Zimmermann. Random matrices with log-range correlations, and log-Sobolev inequalities. Annales mathématiques Blaise Pascal, Tome 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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