  We investigate concentration properties of spectral measures of Hermitian
random matrices with partially dependent entries. More precisely, let $X_n$ be
a Hermitian random matrix of size $n\times n$ that can be split into
independent blocks of the size at most $d_n=o(n^2)$. We prove that under some
mild conditions on the distribution of the entries of $X_n$, the empirical
spectral measure of $X_n$ concentrates around its mean.
  The main theorem is a strengthening of a recent result by Kemp and Zimmerman,
where the size of blocks grows as $o(\log n)$. As an application, we are able
to upgrade the results of Schenker and Schulz on the convergence in expectation
to the semicircle law of a class of random matrices with dependent entries to
weak convergence in probability. Other applications include patterned random
matrices, e.g. matrices of Toeplitz, Hankel or circulant type and matrices with
heavy tailed entries in the domain of attraction of the Gaussian distribution.
