Universality of Wigner Random Matrices

Details Universality of Wigner Random Matrices Universality of Wigner Random Matrices

2009-09-14

arxiv.org/abs/0909.2691v2

Physics

Mathematical Physics

Submitted to the Proceedings of ICMP, Prague, 2009

Scientific paper

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.

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