  In this paper, we derive limit laws for the empirical spectral distributions
of random band and block matrices with correlated entries. In the first part of
the paper, we study band matrices with approximately uncorrelated entries. We
strengthen previously obtained results while requiring weaker assumptions,
which is made possible by a refined application of the method of moments. In
the second part of the paper, we introduce a new two-layered correlation
structure we call SSB-HKW correlated, which enables the study of structured
random matrices with correlated entries. Our results include semicircle laws in
probability and almost surely, but we also obtain other limiting spectral
distributions depending on the conditions. Simple necessary and sufficient
conditions for the limit law to be the semicircle are provided. Our findings
strengthen and extend many results already known.
