RANDOM REVERSIBLE MARKOV MATRICES WITH TUNABLE EXTREMAL EIGENVALUES

• 成果类型: Article
• 署名作者: Chi, Zhiyi
• 署名单位: University of Connecticut
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1146
• 发表日期: 2016
• 页码: 2257-2272
• 关键词: circular law; spectrum

摘要

Random sampling of large Markov matrices with a tunable spectral gap, a nonuniform stationary distribution and a nondegenerate limiting empirical spectral distribution (ESD) is useful. Fix c > 0 and p > 0. Let A(n) be the adjacency matrix of a random graph following G(n, p/n), known as the Erdos-Renyi distribution. Add c/n to each entry of A(n) and then normalize its rows. It is shown that the resulting Markov matrix has the desired properties. Its ESD weakly converges in probability to a symmetric non degenerate distribution, and its extremal eigenvalues, other than 1, fall in [-1/root 1+c/k, -b] boolean OR [b, 1/root 1 + c/k] for any 0 < b < 1/root 1+c, where k = left perpendicularpright perpendicular + 1. Thus, for p is an element of (0, 1), the spectral gap tends to 1-1/root 1+c.