Perturbative and non-perturbative approximation schemes for sparse random matrices
Joseph Baron (University of Bath)
Random matrix theory (RMT) has proved an invaluable tool for the study of systems with disordered interactions in many quite disparate research areas. Widely applicable results, such as the celebrated elliptic law for dense random matrices, allow one to understand when stability is possible in well-connected many-component dynamical systems, for instance. However, such simple and universal results have been more difficult to come by in the case of sparse random matrices, for which the average number of non-zero entries per row (p) is small compared to the matrix size (N). This is despite the fact that sparsity is a common feature of interaction networks in many applications.
In this talk, I will discuss the two ways in which the spectra of sparse random matrices deviate from the classic elliptic law, and I will present approximation schemes that allow us to characterise these novel features analytically. (1) I will show how diagrammatic series can be used to understand the 1/p corrections to the elliptic law. A simple closed-form expression for the « warped ellipse », to which most of the eigenvalues are confined, is derived. (2) I will also demonstrate how a cavity approach can be used to characterise the Lifshitz tails of the spectrum. We will see that eigenvalues in these tail regions of the spectrum correspond to abnormally well-connected hubs of the network (which the random matrix represents), and that the corresponding eigenvectors are exponentially localised around these hubs.