UNIVERSALITY AND THE CIRCULAR LAW FOR SPARSE RANDOM MATRICES

• 成果类型: Article
• 署名作者: Wood, Philip Matchett
• 署名单位: Stanford University
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/11-AAP789
• 发表日期: 2012
• 页码: 1266-1300
• 关键词: eigenvalues; density

摘要

The universality phenomenon asserts that the distribution of the eigenvalues of random matrix with i.i.d. zero mean, unit variance entries does not depend on the underlying structure of the random entries. For example, a plot of the eigenvalues of a random sign matrix, where each entry is +1 or -1 with equal probability, looks the same as an analogous plot of the eigenvalues of a random matrix where each entry is complex Gaussian with zero mean and unit variance. In the current paper, we prove a universality result for sparse random n by n matrices where each entry is nonzero with probability 1/n(1-alpha) where 0 < alpha <= 1 is any constant. One consequence of the sparse universality principle is that the circular law holds for sparse random matrices so long as the entries have zero mean and unit variance, which is the most general result for sparse random matrices to date.