Spiking the random matrix hard edge

Abstract

© 2016, Springer-Verlag Berlin Heidelberg. 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).