Spectra of sparse random matrices

We compute the spectral density for ensembles of of sparse symmetric random matrices using replica. Our formulation of the replica-symmetric ansatz shares the symmetries of the one suggested in a seminal paper by Rodgers and Bray (symmetry with respect to permutation of replica and rotation symmetry in the space of replica), but uses a different representation in terms of superpositions of Gaussians. It gives rise to a pair of integral equations which can be solved by a stochastic population-dynamics algorithm. Remarkably our representation allows us to identify pure-point contributions to the spectral density related to the existence of normalizable eigenstates. Our approach is not restricted to matrices defined on graphs with Poissonian degree distribution. Matrices defined on regular random graphs or on scale-free graphs, are easily handled. We also look at matrices with row constraints such as discrete graph Laplacians. Our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordination and an example of such an unfolding is presented. Our results are well corroborated by numerical diagonalization studies of large finite random matrices.