  A sparse random block matrix model suggested by the Hessian matrix used in
the study of elastic vibrational modes of amorphous solids is presented and
analyzed. By evaluating some moments, benchmarked against numerics, differences
in the eigenvalue spectrum of this model in different limits of space dimension
$d$, and for arbitrary values of the lattice coordination number $Z$, are shown
and discussed. As a function of these two parameters (and their ratio $Z/d$),
the most studied models in random matrix theory (Erdos-Renyi graphs, effective
medium, replicas) can be reproduced in the various limits of block
dimensionality $d$. Remarkably, the Marchenko-Pastur spectral density (which is
recovered by replica calculations for the Laplacian matrix) is reproduced
exactly in the limit of infinite size of the blocks, or $d\rightarrow\infty$,
which for the first time clarifies the physical meaning of space dimension in
these models. The approximate results for $d=3$ provided by our method have
many potential applications in the future, from the vibrational spectrum of
glasses and elastic networks, to wave-localization, disordered conductors,
random resistor networks and random walks.
