We establish eigenvector delocalization and bulk universality for Lévy matrices, which are real, symmetric, $N \times N$ random matrices $\mathbf{H}$ whose upper triangular entries are independent, identically distributed $\alpha$\-stable laws. First, if $\alpha\in(1,2)$ and $E\in\mathbb{R}$ is bounded away from 0, we show that every eigenvector of $\mathbf{H}$ corresponding to an eigenvalue near $E$ is completely delocalized and that the local spectral statistics of $\mathbf{H}$ around $E$ converge to those of the Gaussian Orthogonal Ensemble as $N$ tends to $\infty$. Second, we show for almost all $\alpha\in(0,2)$, there exists a constant $c(\alpha)>0$ such that the same statements hold if $|E|<c(\alpha)$.

莱维矩阵的GOE统计

阿莫尔·阿加瓦尔

帕特里克·洛帕托

姚鸿泽

摘要

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