Special Seminar in Applied Mathematics

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large random matrices in the bulk 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 a potential V and inverse temperature β= 1, 2, 4, corresponding to the orthogonal, unitary and symplectic ensembles. The universality conjecture for invariant ensembles asserts that the local eigenvalue statistics are independent of Vfor all positive real β. In this talk, we review our recent solution to the universality conjecture for both invariant and non-invariant ensembles. The special role played by the logarithmic Sobolev inequality and Dyson Brownian motion will be discussed.