Continuum limits of random matrices and the Brownian carousel

• 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: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0180-z
• 发表日期: 2009
• 页码: 463-508
• 关键词: models

摘要

We show that at any location away from the spectral edge, the eigenvalues of the Gaussian unitary ensemble and its general beta siblings converge to Sine(beta), a translation invariant point process. This process has a geometric description in term of the Brownian carousel, a deterministic function of Brownian motion in the hyperbolic plane. The Brownian carousel, a description of the a continuum limit of random matrices, provides a convenient way to analyze the limiting point processes. We show that the gap probability of Sine(beta) is continuous in the gap size and beta, and compute its asymptotics for large gaps. Moreover, the stochastic differential equation version of the Brownian carousel exhibits a phase transition at beta = 2. [GRAPHICS]