Levy-Khintchine random matrices and the Poisson weighted infinite skeleton tree

We study a class of Hermitian random matrices which includes and generalizes Wigner matrices, heavy-tailed random matrices, and sparse random matrices such as the adjacency matrices of Erdos-Renyi random graphs with p ~ 1/N. Our NxN random matrices have real entries which are i.i.d. up to symmetry. The distributions may depend on N, however, the sums of rows must converge in distribution; it is then well-known that the limiting distributions are infinitely divisible. We show that a limiting empirical spectral distribution (LSD) exists, and via local weak convergence of associated graphs, the LSD corresponds to the spectral measure at the root of a Poisson weighted infinite skeleton tree. This graph is formed by connecting infinitely many Poisson weighted infinite trees using a backbone structure. One example covered by the results are matrices with i.i.d. entries having infinite second moments, but normalized to be in the Gaussian domain of attraction. In this case, the limiting graph is just the positive integer line rooted at 1, and as expected, the LSD is Wigner's semi-circle law. The results also extend to self-adjoint complex matrices and also to Wishart matrices.