WHAT IS THE PROBABILITY THAT A LARGE RANDOM MATRIX HAS NO REAL EIGENVALUES?
成果类型：
Article
署名作者：
Kanzieper, Eugene; Poplavskyi, Mihail; Timm, Carsten; Tribe, Roger; Zaboronski, Oleg
署名单位：
Weizmann Institute of Science; University of Warwick; Technische Universitat Dresden
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/15-AAP1160
发表日期：
2016
页码：
2733-2753
关键词：
摘要：
We study the large-n limit of the probability P-2n,P-2k that a random 2n x 2n matrix sampled from the real Ginibre ensemble has 2k real eigenvalues. We prove that lim(n ->infinity)1/root 2n log P-2n,P-2k = lim(n ->infinity) 1/root 2n log P-2n,P-0 = -1/root 2 pi zeta (3/2), where zeta is the Riemann zeta-function. Moreover, for any sequence of non-negative integers (k(n))(n >= 1), lim(n ->infinity) 1/root 2n log p(2n,2kn) = -1/root 2 pi zeta (3/2), provided lim(n ->infinity) (n(-1/2) log(n))k(n) = 0.