  We study the large $N$ limit of a sparse random block matrix ensemble. It
depends on two parameters: the average connectivity $Z$ and the size of the
blocks $d$, which is the dimension of an euclidean space.
  In the limit of large $d$, with $\frac{Z}{d}$ fixed, we prove the conjecture
that the spectral distribution of the sparse random block matrix converges in
the case of the Adjacency block matrix to the one of the effective medium
approximation, in the case of the Laplacian block matrix to the
Marchenko-Pastur distribution.
  We extend previous analytical computations of the moments of the spectral
density of the Adjacency block matrix and the Lagrangian block matrix, valid
for all values of $Z$ and $d$.
