The Annals of Probability

Limits of spiked random matrices II

Alex Bloemendal and Bálint Virág

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Abstract

The top eigenvalues of rank $r$ spiked real Wishart matrices and additively perturbed Gaussian orthogonal ensembles are known to exhibit a phase transition in the large size limit. We show that they have limiting distributions for near-critical perturbations, fully resolving the conjecture of Baik, Ben Arous and Péché [Duke Math. J. (2006) 133 205–235]. The starting point is a new $(2r+1)$-diagonal form that is algebraically natural to the problem; for both models it converges to a certain random Schrödinger operator on the half-line with $r\times r$ matrix-valued potential. The perturbation determines the boundary condition and the low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. We treat the real, complex and quaternion ($\beta=1,2,4$) cases simultaneously. We further characterize the limit laws in terms of a diffusion related to Dyson’s Brownian motion, or alternatively a linear parabolic PDE; here $\beta$ appears simply as a parameter. At $\beta=2$, the PDE appears to reconcile with known Painlevé formulas for these $r$-parameter deformations of the GUE Tracy–Widom law.

Article information

Source
Ann. Probab., Volume 44, Number 4 (2016), 2726-2769.

Dates
Received: June 2012
Revised: May 2015
First available in Project Euclid: 2 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1470139153

Digital Object Identifier
doi:10.1214/15-AOP1033

Mathematical Reviews number (MathSciNet)
MR3531679

Zentralblatt MATH identifier
06631783

Subjects
Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)

Keywords
Random matrix theory finite rank perturbations spiked model Tracy–Widom distributions BBP phase transition stochastic Airy operator

Citation

Bloemendal, Alex; Virág, Bálint. Limits of spiked random matrices II. Ann. Probab. 44 (2016), no. 4, 2726--2769. doi:10.1214/15-AOP1033. https://projecteuclid.org/euclid.aop/1470139153


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