EMPIRICAL DISTRIBUTIONS OF LAPLACIAN MATRICES OF LARGE DILUTE RANDOM GRAPHS

被引：12

作者：

Jiang, Tiefeng ^{[1
]}

机构：

[1] Univ Minnesota, Sch Stat, 224 Church St, Minneapolis, MN 55455 USA

来源：

RANDOM MATRICES-THEORY AND APPLICATIONS | 2012年 / 1卷 / 03期

基金：

美国国家科学基金会;

关键词：

Random matrix; random graph; dilute graph; Laplacian matrix; normalized Laplacian matrix; spectral distribution; semi-circle law; free convolution; EIGENVALUE DISTRIBUTION;

D O I：

10.1142/S2010326312500049

中图分类号：

O4 [物理学];

学科分类号：

0702 ;

摘要：

We study the spectral properties of the Laplacian matrices and the normalized Laplacian matrices of the Erdos-Renyi random graph G(n, p(n)) for large n. Although the graph is simple, we discover some interesting behaviors of the two Laplacian matrices. In fact, under the dilute case, that is, p(n) is an element of (0, 1) and np(n) -> infinity, we prove that the empirical distribution of the eigenvalues of the Laplacian matrix converges to a deterministic distribution, which is the free convolution of the semi-circle law and N(0, 1). However, for its normalized version, we prove that the empirical distribution converges to the semi-circle law.

引用

页数：20