Random Matrices with Log-Range Correlations, and Log-Sobolev Inequalities

Todd Kemp, David Zimmermann

Published 2014-05-11, updated 2016-12-06Version 2

Let $X_N$ be a symmetric $N\times N$ random matrix whose $\sqrt{N}$-scaled centered 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 $X_N$ converges weakly to its mean in probability. 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 prove this result be developing a new log-Sobolev inequality, generalizing the first 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 $\mathbf{Y}+t\mathbf{Z}$ satisfies a log-Sobolev inequality for all $t>0$, and we give bounds on the optimal log-Sobolev constant.