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Spiking the random matrix hard edge

2016; Springer Science+Business Media; Volume: 169; Issue: 1-2 Linguagem: Inglês

10.1007/s00440-016-0733-1

1432-2064

José A. Ramı́rez, Brian Rider,

Stochastic processes and statistical mechanics

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 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 Peché (Ann Probab 33:1643–1697, 2005), and from a unified $$\beta $$ random operator point of view by Bloemendal and Virág (Probab Theory Relat Fields 156:795–825, 2013; Ann Probab arXiv:1109.3704 , 2011).