Bourgade, PaulErdos, LaszloYau, Horng-TzerYin, Jun2022-12-212015Bourgade, Paul, Laszlo Erdős, Horng-Tzer Yau, and Jun Yin. 2015. “Fixed Energy Universality for Generalized Wigner Matrices.” Communications on Pure and Applied Mathematics 69 (10) (December 15): 1815–1881. doi:10.1002/cpa.21624.0010-3640https://nrs.harvard.edu/URN-3:HUL.INSTREPOS:37373919We 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.en-USJournal Article2022-12-2110.1002/cpa.21624