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arXiv 1001.3453math-phmath.MPmath.PR
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广义Wigner矩阵的体普适性

Bulk universality for generalized Wigner matrices

Laszlo Erdos, Horng-Tzer Yau, Jun Yin

备注：
Appendix B is simplified; an extra Assumption IV was added to Thm 6.2. On Sep 17, 2011 a small error in the conditions of Lemma 7.8 was fixed and the proof of Lemma 7.5 in pages 38-39 adjusted
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摘要：

考虑$N\times N$厄密随机矩阵或对称随机矩阵$H$，其中$(i,j)$矩阵元素的分布由具有次指数衰减的概率测度$ν_{ij}$给出。令$σ_{ij}^2$为概率度量$ν_{ij}$的方差，其归一化属性为所有$j$的$\sum_{i} σ^2_{ij} = 1$。在基本上唯一的条件下，$c\le N σ_{ij}^2 \le c^{-1}$为一些常数$c>0$，我们证明了，在限制$N \to \infty$，特征值间距统计$H$在大部分的频谱与高斯酉或正交系综（GUE或GOE）。我们还表明，对于带宽$M$的带矩阵的本地循环法持有的能量标度$M^{-1}$。

英文摘要：

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 $ν_{ij}$ with a subexponential decay. Let $σ_{ij}^2$ be the variance for the probability measure $ν_{ij}$ with the normalization property that $\sum_{i} σ^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N σ_{ij}^2 \le c^{-1}$ 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 $M^{-1}$.

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