Spiking the random matrix hard edge

Abstract

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 beta ensembles. For multiple spikes, the necessary construction restricts matters to real, complex or quaternion (beta=1, 2, or 4) ensembles. The limit laws are described in terms of a 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 beta=2 by Baik, Ben Arous, and Peche, and from a unified beta random operator point of view by Bloemendal and Virag.