Fixed energy universality for generalized Wigner matrices

Abstract

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 motion and show that microscopic universality follows from mesoscopic statistics.

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Orthogonal and symplectic matrix models: universality and other properties

Cited by in corpus (12)

• Bulk universality of sparse random matrices
• Fractional Brownian motion with Hurst index $H=0$ and the Gaussian Unitary Ensemble
• Local Kesten--McKay law for random regular graphs
• Bulk eigenvalue statistics for random regular graphs
• Gaussian and non-Gaussian fluctuations for mesoscopic linear statistics in determinantal processes
• Mesoscopic linear statistics of Wigner matrices
• Bulk universality for random lozenge tilings near straight boundaries and for tensor products
• Random matrices: tail bounds for gaps between eigenvalues
• Mesoscopic linear statistics of Wigner matrices of mixed symmetry class
• Random matrices, log-gases and Holder regularity
• Universality in several-matrix models via approximate transport maps
• Overcrowding asymptotics for the Sine_beta process