Bulk universality of sparse random matrices

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...

Typical l1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices

Typical l 1-recovery limit of sparse vectors represented by concatenations of random orthogonal matrices Abstract. We consider the problem of recovering an N-dimensional sparse vector x from its linear transformation y = Dx of M (< N) dimension. Minimizing the l 1-norm of x under the constraint y = Dx is a standard approach for the recovery problem, and earlier studies report that the critical ...

Universality in the bulk for random matrices Notes for Lectures

Universality Limits for Random Matrices and de Branges Spaces of Entire Functions

We prove that de Branges spaces of entire functions describe universality limits in the bulk for random matrices, in the unitary case. In particular, under mild conditions on a measure with compact support, we show that each possible universality limit is the reproducing kernel of a de Branges space of entire functions that equals a classical Paley-Wiener space. We also show that any such repro...

Fixed energy universality for generalized Wigner matrices

We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motio...