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Abstract: We consider $N\times N$ Hermitian Wigner random matrices $H$ where theprobability density for each matrix element is given by the density $ ux=e^{- Ux}$. We prove that the eigenvalue statistics in the bulk is given byDyson sine kernel provided that $U \in C^6\RR$ with at most polynomiallygrowing derivatives and $ ux \le C e^{- C |x|}$ for $x$ large. The proof isbased upon an approximate time reversal of the Dyson Brownian motion combinedwith the convergence of the eigenvalue density to the Wigner semicircle law onshort scales.



Author: Laszlo Erdos, Sandrine Peche, Jose A. Ramirez, Benjamin Schlein, Horng-Tzer Yau

Source: https://arxiv.org/



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