Cavity and replica methods for the spectral density of sparse symmetric random matrices

We review the problem of how to compute spectral density sparse symmetric random matrices, i.e. weighted adjacency matrices undirected graphs. Starting from Edwards-Jones formula, we illustrate milestones this line research, including pioneering work Bray and Rodgers using replicas. focus first on cavity method, showing that it quickly provides correct recursion equations both for single instances at ensemble level. also describe an alternative replica solution proves be equivalent method. Both derivations allow us obtain via integral equation auxiliary probability function. show can solved a stochastic population dynamics algorithm, provide its implementation. In formalism, is naturally written in terms superposition local contributions nodes given degree, whose role thoroughly elucidated. This paper does not contain original material, but rather gives pedagogical overview topic. It indeed addressed students researchers who consider entering field. theoretical tools numerical algorithms are discussed detail, highlighting conceptual subtleties practical aspects.