  The Wigner-Gaudin-Mehta-Dyson conjecture asserts that the local eigenvalue
statistics of large random matrices exhibit universal behavior depending only
on the symmetry class of the matrix ensemble. For invariant matrix models, the
eigenvalue distributions are given by a log-gas with inverse temperature $\beta
= 1, 2, 4$, corresponding to the orthogonal, unitary and symplectic ensembles.
For $\beta \not \in \{1, 2, 4\}$, there is no matrix model behind this model,
but the statistical physics interpretation of the log-gas is still valid for
all $\beta > 0$. The universality conjecture for invariant ensembles asserts
that the local eigenvalue statistics are independent of $V$. In this article,
we review our recent solution to the universality conjecture for both invariant
and non-invariant ensembles. We will also demonstrate that the local ergodicity
of the Dyson Brownian motion is the intrinsic mechanism behind the
universality. Furthermore, we review the solution of Dyson's conjecture on the
local relaxation time of the Dyson Brownian motion. Related questions such as
delocalization of eigenvectors and local version of Wigner's semicircle law
will also be discussed.
