Bulk universality for generalized Wigner matrices

Bulk universality for generalized Wigner matrices,Laszlo Erdos,Horng-Tzer Yau,Jun Yin

Bulk universality for generalized Wigner matrices   (Citations: 9)
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Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$ with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.
Published in 2010.
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