  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.
