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

The remarkable universality of the eigenvalue correlation functions is perhaps one of the most salient observations 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 random matrix ensemble. In this work, we show that, in contrast, the long-range correlations of the eigenvalues of random matrices are more sensitive. We find that for sparse matrices, matrices with non-Gaussian statistics, and matrices with statistical heterogeneity, long-range correlations emerge, which give rise to a non-zero eigenvalue compressibility. This behaviour has previously been linked to the presence of non-ergodic extended eigenvectors and multi-fractality. We provide succinct analytical expressions for the two-point Green's functions and eigenvalue correlations using a perturbative diagrammatic approach. We discuss how these results also relate to recent findings on the large deviation functions of the eigenvalues and the breakdown of simple mean-field theories in disordered systems.