STOCHASTIC AIRY SEMIGROUP THROUGH TRIDIAGONAL MATRICES

• 成果类型: Article
• 署名作者: Gorin, Vadim; Shkolnikov, Mykhaylo
• 署名单位: Massachusetts Institute of Technology (MIT); Kharkevich Institute for Information Transmission Problems of the RAS; Russian Academy of Sciences; Princeton University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1229
• 发表日期: 2018
• 页码: 2287-2344
• 关键词: universality; spectrum; ensembles; edge; fluctuations; eigenvalues; models

摘要

We determine the operator limit for large powers of random symmetric tridiagonal matrices as the size of the matrix grows. The result provides a novel expression in terms of functionals of Brownian motions for the Laplace transform of the Airy beta process, which describes the largest eigen-values in the beta ensembles of random matrix theory. Another consequence is a Feynman-Kac formula for the stochastic Airy operator of Edelman-Sutton and Ramirez-Rider-Virag. As a side result, we find that the difference between the area underneath a standard Brownian excursion and one half of the integral of its squared local times is a Gaussian random variable.