Spiking the random matrix hard edge

• Published in: Probability theory and related fields Vol. 169; no. 1-2; pp. 425 - 467
• Main Authors: Ramírez, José A., Rider, Brian
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2017; Springer Nature B.V
• Subjects: Covariance; Deformation; Differential equations; Distribution functions; Economics; Finance; Insurance; Management; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Operators (mathematics); Partial differential equations; Probability; Probability Theory and Stochastic Processes; Quantitative Finance; Quaternions; Statistics for Business; Theoretical; 60B20; 60F05
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-016-0733-1

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 β ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion ( β = 1 , 2 , 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 β = 2 by Baik, Ben Arous, and Peché (Ann Probab 33:1643–1697, 2005 ), and from a unified β random operator point of view by Bloemendal and Virág (Probab Theory Relat Fields 156:795–825, 2013 ; Ann Probab arXiv:1109.3704 , 2011 ).