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arXiv · 0905.4176

Bulk Universality for Wigner Matrices

Laszlo Erdos·Sandrine Peche·Jose A. Ramirez·Benjamin Schlein·Horng-Tzer Yau
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Abstract

We consider $N\times N$ Hermitian Wigner random matrices $H$ where the probability density for each matrix element is given by the density $ν(x)= e^{- U(x)}$. We prove that the eigenvalue statistics in the bulk is given by Dyson sine kernel provided that $U \in C^6(\RR)$ with at most polynomially growing derivatives and $ν(x) \le C e^{- C |x|}$ for $x$ large. The proof is based upon an approximate time reversal of the Dyson Brownian motion combined with the convergence of the eigenvalue density to the Wigner semicircle law on short scales.

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Laszlo Erdos, Sandrine Peche, Jose A. Ramirez, Benjamin Schlein, Horng-Tzer Yau. 2009-10-20. Bulk Universality for Wigner Matrices. https://arxiv.org/abs/0905.4176

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