  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.
