Abstract
We review the problem of how to compute the spectral density of sparse symmetric random matrices, i.e. weighted adjacency matrices of undirected graphs. Starting from the Edwards-Jones formula, we illustrate the milestones of this line of research, including the pioneering work of Bray and Rodgers using replicas. We focus first on the cavity method, showing that it quickly provides the correct recursion equations both for single instances and at the ensemble level. We also describe an alternative replica solution that proves to be equivalent to the cavity method. Both the cavity and the replica derivations allow us to obtain the spectral density via the solution of an integral equation for an auxiliary probability density function. We show that this equation can be solved using a stochastic population dynamics algorithm, and we provide its implementation. In this formalism, the spectral density is naturally written in terms of a superposition of local contributions from nodes of given degree, whose role is thoroughly elucidated. This paper does not contain original material, but rather gives a pedagogical overview of the topic. It is indeed addressed to students and researchers who consider entering the field. Both the theoretical tools and the numerical algorithms are discussed in detail, highlighting conceptual subtleties and practical aspects.
Highlights
The calculation of the average spectral density of eigenvalues of random matrices belonging to a certain ensemble has traditionally been one the fundamental problems in Random Matrix Theory (RMT), ever since the application of RMT to the statistics of energy levels of heavy nuclei [1]
As an illustration of the fact that the formalism presented here can be used to obtain the spectral density for other finite mean degree ensembles in the configuration model class, we show in right panel of Fig. 5 the spectral density of the ensemble of adjacency matrices of random regular graphs (RRG), having degree distribution p(k) = δk,c
We started with the celebrated Edwards-Jones formula (2) and outlined its proof
Summary
The calculation of the average spectral density of eigenvalues of random matrices belonging to a certain ensemble has traditionally been one the fundamental problems in Random Matrix Theory (RMT), ever since the application of RMT to the statistics of energy levels of heavy nuclei [1]. Tightbinding Hamiltonian operators with a kinetic term and an on-site random potential translate into matrix models that involve discrete graph Laplacians with additional random contributions to diagonals [15]. The spectra of such matrices have been used for the characterisation of many physical systems in condensed matter such as the study of gelation transitions in polymers [16]. Our analysis is rooted in the statistical mechanics of disordered systems, with the main technical tools being the cavity (Section 3) and replica methods (Section 4 and 5)
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