Universality for 1d Random Band Matrices: Sigma-Model Approximation

Universality for a class of random band matrices

We prove the universality for the eigenvalue gap statistics in the bulk of the spectrum for band matrices, in the regime where the band width is comparable with the dimension of the matrix, W ∼ N . All previous results concerning universality of non-Gaussian random matrices are for meanfield models. By relying on a new mean-field reduction technique, we deduce universality from quantum unique e...

Universality of Wigner Random Matrices

We consider N×N symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure ν with a subexponential decay. We prove that the local eigenvalue statistics in the bulk of the spectrum for these matrices coincide with those of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary ...

Bulk universality of sparse random matrices

The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters. We consider the adjacency matrix of the ensemble of Erd˝ os-Rényi random graphs which consists of graphs on N vertices in which each edge occurs independently with probability p. We prove that in the regime pN ≫ 1, these matrices exhibit bulk universality in the sense tha...

Analytical Results for Random Band Matrices with

Using the supersymmetry method we analytically calculate the local density of states, the localiztion length, the generalized inverse participation ratios , and the distribution function of eigenvector components for the superpo-sition of a random band matrix with a strongly uctuating diagonal matrix. In this way we extend previously known results for ordinary band matrices to the class of rand...

Universality in the bulk for random matrices Notes for Lectures