Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory

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 $d$.