Spiking the random matrix hard edge

• 成果类型: Article
• 署名作者: Ramirez, Jose A.; Rider, Brian
• 署名单位: Universidad Costa Rica; Pennsylvania Commonwealth System of Higher Education (PCSHE); Temple University
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0733-1
• 发表日期: 2017
• 页码: 425-467
• 关键词: level-spacing distributions; beta ensembles; EIGENVALUE; TRANSITION; kernel

摘要

We characterize the limiting smallest eigenvalue distributions (or hard edge laws) for sample covariance type matrices drawn from a spiked population. In the case of a single spike, the results are valid in the context of the general ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion ( or 4) ensembles. The limit laws are described in terms of random integral operators, and partial differential equations satisfied by the corresponding distribution functions are derived as corollaries. We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy-Widom laws). The latter were first described at by Baik, Ben Arous, and Pech, (Ann Probab 33:1643-1697, 2005), and from a unified random operator point of view by Bloemendal and Virag (Probab Theory Relat Fields 156:795-825, 2013; Ann Probab arXiv:1109.3704, 2011).