Limits of spiked random matrices I

• 成果类型: Article
• 署名作者: Bloemendal, Alex; Virag, Balint
• 署名单位: Harvard University; University of Toronto; University of Toronto
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-012-0443-2
• 发表日期: 2013
• 页码: 795-825
• 关键词: TRACY-WIDOM LIMIT; LARGEST EIGENVALUE; distributions; asymptotics; UNIVERSALITY; fluctuations; spectrum; THEOREM; models

摘要

Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank one spiked real Wishart setting and its general beta analogue, proving a conjecture of Baik et al. (Ann Probab 33:1643-1697, 2005). We also treat shifted mean Gaussian orthogonal and beta ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schrodinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which beta appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at beta = 2,4, yielding in particular a new and simple proof of the Painlev, representations for these Tracy-Widom distributions.