Classes of non-Gaussian random matrices: long-range eigenvalue correlations and non-ergodic extended eigenvectors

Abstract

The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient findings in random matrix theory. Particularly for short-range separations of the eigenvalues, the correlation functions have been shown to be robust to many changes in the random matrix ensemble, and are often well-predicted by results corresponding to Gaussian random matrices in many applications. In this work, we show that, in contrast, the long-range correlations of the eigenvalues of random matrices are more sensitive. Using a path-integral approach, we identify classes of statistical deviations from the Gaussian Orthogonal random matrix Ensemble (GOE) that give rise to long-range correlations. We provide closed-form analytical expressions for the eigenvalue compressibility and two-point correlations, which ordinarily vanish for the GOE, but are non-zero here. These expressions are universal in the limit of small non-Gaussianity. We discuss how these results suggest the presence of non-ergodic eigenvectors, and we verify numerically that the eigenvector component distributions of a wide variety of non-Gaussian ensembles exhibit the associated power-law tails. We also comment on how these findings reveal the need to go beyond simple mean-field theories in disordered systems with non-Gaussian interactions.