Wigner随机矩阵的Wegner估计和水平排斥
Wegner estimate and level repulsion for Wigner random matrices
摘要：
考虑元素独立同分布的$N\times N$厄米特随机矩阵（Wigner矩阵）.对矩阵进行归一化，使得连续特征值之间的平均间距为$1/N$阶。在对单个矩阵元分布的适当假设下，我们首先证明了在远离谱边缘的地方，本征值的经验密度集中在能量标度$η\gg N^{-1}$上的Wigner周期律附近.这个结果建立了最优尺度上的对数律，并且从我们以前的结果\cite{ESY2}中删除了一个对数因子。然后，我们展示了一个Wegner估计，即平均态密度是有界的。最后，我们证明了Wigner矩阵的特征值是相互排斥的，与普适性猜想一致。
英文摘要：
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. Under suitable assumptions on the distribution of the single matrix element, we first prove that, away from the spectral edges, the empirical density of eigenvalues concentrates around the Wigner semicircle law on energy scales $η\gg N^{-1}$. This result establishes the semicircle law on the optimal scale and it removes a logarithmic factor from our previous result \cite{ESY2}. We then show a Wegner estimate, i.e. that the averaged density of states is bounded. Finally, we prove that the eigenvalues of a Wigner matrix repel each other, in agreement with the universality conjecture.
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