  Given an ensemble of NxN random matrices, a natural question to ask is
whether or not the empirical spectral measures of typical matrices converge to
a limiting spectral measure as N --> oo. While this has been proved for many
thin patterned ensembles sitting inside all real symmetric matrices, frequently
there is no nice closed form expression for the limiting measure. Further,
current theorems provide few pictures of transitions between ensembles. We
consider the ensemble of symmetric m-block circulant matrices with entries
i.i.d.r.v. These matrices have toroidal diagonals periodic of period m. We view
m as a "dial" we can "turn" from the thin ensemble of symmetric circulant
matrices, whose limiting eigenvalue density is a Gaussian, to all real
symmetric matrices, whose limiting eigenvalue density is a semi-circle. The
limiting eigenvalue densities f_m show a visually stunning convergence to the
semi-circle as m tends to infinity, which we prove. In contrast to most studies
of patterned matrix ensembles, our paper gives explicit closed form expressions
for the densities. We prove that f_m is the product of a Gaussian and a degree
2m-2 polynomial; the formula equals that of the m x m Gaussian Unitary Ensemble
(GUE). The proof is by the moments. The new feature, which allows us to obtain
closed form expressions, is converting the central combinatorial problem in the
moment calculation into an equivalent counting problem in algebraic topology.
We end with a generalization of the m-block circulant pattern, dropping the
assumption that the m random variables be distinct. We prove that the limiting
spectral distribution exists and is determined by the pattern of the
independent elements within an m-period, depending on not only the frequency at
which each element appears, but also the way the elements are arranged.
