Joseph Baron

Random matrix non-Gaussianity as the origin of long-range eigenvalue correlations

Random matrix theory's (RMT's) broad aim is to understand the properties of large matrices in terms of the statistics of their elements. Perhaps some of the most impactful results from RMT concern the correlations between eigenvalues, which have had tremendous success in predicting the level spacing distributions in heavy nuclei and chaotic quantum systems, for example. This success owes to the remarkable universality of the correlations of near-by eigenvalues. That is, the short-range eigenvalue correlations are robust to many changes in the random matrix ensemble. In this talk, I will discuss how the long-range correlations of eigenvalues are more malleable. Using a field-theoretic approach and Feynman diagrams, I will show how one can identify the classes of random matrix ensemble that give rise to long-range eigenvalue correlations, which are absent in Gaussian random matrices. Examples include the adjacency matrices of sparse or highly heterogeneous networks. Long-range correlations act as a diagnostic test for the presence of non-ergodic extended eigenvectors and multi-fractality, which have indeed previously been observed in such networks.