论文标题
Lévy矩阵的特征向量统计
Eigenvector Statistics of Lévy Matrices
论文作者
论文摘要
我们分析了重尾随机对称矩阵（也称为Lévy矩阵）的特征向量入口的统计数据，其相关特征值足够小。我们表明，任何此类条目的限制定律都是非高斯的，由正态分布的乘积带有另一个随机变量，该变量取决于相应特征值的位置。尽管后一个随机变量通常是不明显的，但对于中位特征向量，它是由单方面稳定定律的倒数给出的。此外，我们表明，同一特征向量的不同条目在渐近独立上，但是特征向量与附近特征值之间存在非平凡的相关性。我们的发现与Wigner矩阵和稀疏随机图的已知特征向量行为形成鲜明对比。
We analyze statistics for eigenvector entries of heavy-tailed random symmetric matrices (also called Lévy matrices) whose associated eigenvalues are sufficiently small. We show that the limiting law of any such entry is non-Gaussian, given by the product of a normal distribution with another random variable that depends on the location of the corresponding eigenvalue. Although the latter random variable is typically non-explicit, for the median eigenvector it is given by the inverse of a one-sided stable law. Moreover, we show that different entries of the same eigenvector are asymptotically independent, but that there are nontrivial correlations between eigenvectors with nearby eigenvalues. Our findings contrast sharply with the known eigenvector behavior for Wigner matrices and sparse random graphs.
