Random matrices, log-gases and Holder regularity

The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gases that represent a universal model for strongly correlated systems. Finally, in the spirit of Wigner's original vision, we discuss the extensions of these universality results to more realistic physical systems such as random band matrices.Comment: Proceedings of ICM 201

Universality of Wigner Random Matrices

We consider $N\times N$ symmetric or hermitian random matrices with independent, identically distributed entries where the probability distribution for each matrix element is given by a measure $\nu$ with a subexponential decay. We prove that the local eigenvalue statistics in the bulk of the spectrum for these matrices coincide with those of the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Unitary Ensemble (GUE), respectively, in the limit $N\to \infty$. Our approach is based on the study of the Dyson Brownian motion via a related new dynamics, the local relaxation flow. We also show that the Wigner semicircle law holds locally on the smallest possible scales and we prove that eigenvectors are fully delocalized and eigenvalues repel each other on arbitrarily small scales.Comment: Submitted to the Proceedings of ICMP, Prague, 200

Recent developments in quantum mechanics with magnetic fields

We present a review on the recent developments concerning rigorous mathematical results on Schr\"odinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon.Comment: Update of the previous versions; some more references added and typos and some minor errors correcte

The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case

We consider the spectral statistics of large random band matrices on mesoscopic energy scales. We show that the two-point correlation function of the local eigenvalue density exhibits a universal power law behaviour that differs from the Wigner-Dyson-Mehta statistics. This law had been predicted in the physics literature by Altshuler and Shklovskii [4]; it describes the correlations of the eigenvalue density in general metallic samples with weak disorder. Our result rigorously establishes the Altshuler-Shklovskii formulas for band matrices. In two dimensions, where the leading term vanishes owing to an algebraic cancellation, we identify the first non-vanishing term and show that it differs substantially from the prediction of Kravtsov and Lerner [33]