Fixed energy universality for generalized Wigner matrices

Understand

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.