# Mathematical Sciences Research Institute

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# Seminar

Limits of spiked random matrices December 01, 2010
Parent Program: -- MSRI: Simons Auditorium
Description Speaker: Alex Bloemendal

Abstract:  Given a large, high-dimensional sample from a spiked
population, the top sample covariance eigenvalue is known to exhibit a phase
transition. We show that the largest eigenvalues have asymptotic
distributions near the phase transition in the rank one spiked real
Wishart setting and its general beta analogue, proving a conjecture of
Baik, Ben Arous and Péché.  One obtains the known limiting random
Schrödinger operator on the half-line, but the boundary condition now
depends on the perturbation. We derive a characterization of the limit
laws in terms of a simple linear boundary value problem in which beta
appears as a parameter. This PDE description recovers known explicit
formulas at beta=2,4, yielding in particular a new proof of the
Painlevé representations for these Tracy-Widom distributions.  If time
permits we will discuss the generalization to the higher-rank spiked
model.  This work is joint with Bálint Virág.
Video