LARGE GAPS BETWEEN RANDOM EIGENVALUES

• 成果类型: Article
• 署名作者: Valko, Benedek; Virag, Balint
• 署名单位: University of Wisconsin System; University of Wisconsin Madison; University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP508
• 发表日期: 2010
• 页码: 1263-1279
• 关键词: Asymptotics; constant

摘要

We show that in the point process limit of the bolt eigenvalues of beta-ensembles of random matrices, the probability of having no eigenvalue ill a fixed interval of size lambda is given by (kappa beta + sigma(1))lambda(gamma beta) exp(-beta/64 lambda(2) + (beta/8-1/4)lambda) as lambda --> infinity, where gamma beta = 1/4(beta/2 + 2/beta -3) and kappa beta is an undetermined positive constant. This is a slightly corrected version of a prediction by Dyson 11 Math. Phys. 3 (1962) 157-1651. Our proof uses the new Brownian carousel representation of the limit process. as well as the Cameron Martin Girsanov transformation in stochastic calculus.