论文标题
重尾椭圆随机矩阵的光谱
Spectrum of Heavy-Tailed Elliptic Random Matrices
论文作者
论文摘要
椭圆随机矩阵$ x $是一个方形矩阵，其$（i，j）$ - 条目$ x_ {ij} $独立于其余条目，除了可能$ x_ {ji} $。椭圆形随机矩阵概括了Wigner矩阵和具有独立条目的非热矩阵。当椭圆形随机矩阵的条目平均为零和单位方差时，已知经验光谱分布会收敛到由镜像条目的协方差确定的椭圆内部的均匀分布。 我们认为椭圆形的随机矩阵的条目没有两个有限的时刻。我们的主要结果表明，当椭圆形随机矩阵的条目处于吸引$α$稳定的随机变量的域时，$ 0 <α<2 $的范围时，经验光谱测量的可能性就会收敛到确定性极限。这概括了Bordenave，Caputo和Chafaï的结果，这些矩阵具有独立且分布相同的条目。证明的关键要素是（i）在任何时刻假设下椭圆形随机矩阵最小的奇异值的一般界限； （ii）从适当的意义上讲，矩阵与泊松加权无限树上随机操作员的收敛性。
An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is independent of the rest of the entries except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with independent entries. When the entries of an elliptic random matrix have mean zero and unit variance, the empirical spectral distribution is known to converge to the uniform distribution on the interior of an ellipse determined by the covariance of the mirrored entries. We consider elliptic random matrices whose entries fail to have two finite moments. Our main result shows that when the entries of an elliptic random matrix are in the domain of attraction of an $α$-stable random variable, for $0<α<2$, the empirical spectral measure converges, in probability, to a deterministic limit. This generalizes a result of Bordenave, Caputo, and Chafaï for heavy-tailed matrices with independent and identically distributed entries. The key elements of the proof are (i) a general bound on the least singular value of elliptic random matrices under no moment assumptions; and (ii) the convergence, in an appropriate sense, of the matrices to a random operator on the Poisson Weighted Infinite Tree.
