论文标题
Lévy-khintchine随机拉普拉斯矩阵的光谱
Spectrum of Lévy-Khintchine Random Laplacian Matrices
论文作者
论文摘要
我们考虑表格$ l_n = a_n-d_n $的随机拉普拉斯矩阵的频谱，其中$ a_n $是一个真正的对称随机矩阵，$ d_n $是对角矩阵，其条目等于$ a_n $的相应行总和。如果$ a_n $是一个带有高斯分布域中的条目的Wigner矩阵，则已知$ l_n $的经验光谱度量将融合到半圆形分布的自由卷积和标准的真实高斯分布。 我们认为真实的对称随机矩阵$ a_n $具有独立条目（直至对称），其行总和将纯粹的非高斯无限分配分布收敛，该分布属于lévy-khintchine随机矩阵的类别，该矩阵首先引入了由Jung First介绍的[Trans Am Math Soc，\ Trans Am Math Soc，\ textbf，\ textbf {370}，（2018}，（2018年），（2018年）。我们的主要结果表明，$ l_n $的经验光谱度量几乎可以肯定地收敛到确定性限制。证明的关键步骤是使用行总和纯粹的非高斯性质来构建一个随机操作员，$ l_n $从适当意义上收集到该操作员。该操作员导致递归分布方程式独特地描述了极限经验光谱测量的stieltjes变换。
We consider the spectrum of random Laplacian matrices of the form $L_n=A_n-D_n$ where $A_n$ is a real symmetric random matrix and $D_n$ is a diagonal matrix whose entries are equal to the corresponding row sums of $A_n$. If $A_n$ is a Wigner matrix with entries in the domain of attraction of a Gaussian distribution the empirical spectral measure of $L_n$ is known to converge to the free convolution of a semicircle distribution and a standard real Gaussian distribution. We consider real symmetric random matrices $A_n$ with independent entries (up to symmetry) whose row sums converge to a purely non-Gaussian infinitely divisible distribution, which fall into the class of Lévy-Khintchine random matrices first introduced by Jung [Trans Am Math Soc, \textbf{370}, (2018)]. Our main result shows that the empirical spectral measure of $L_n$ converges almost surely to a deterministic limit. A key step in the proof is to use the purely non-Gaussian nature of the row sums to build a random operator to which $L_n$ converges in an appropriate sense. This operator leads to a recursive distributional equation uniquely describing the Stieltjes transform of the limiting empirical spectral measure.
