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July 2016 Limits of spiked random matrices II
Alex Bloemendal, Bálint Virág
Ann. Probab. 44(4): 2726-2769 (July 2016). DOI: 10.1214/15-AOP1033


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.


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Alex Bloemendal. Bálint Virág. "Limits of spiked random matrices II." Ann. Probab. 44 (4) 2726 - 2769, July 2016.


Received: 1 June 2012; Revised: 1 May 2015; Published: July 2016
First available in Project Euclid: 2 August 2016

zbMATH: 06631783
MathSciNet: MR3531679
Digital Object Identifier: 10.1214/15-AOP1033

Primary: 60B12 , 60B20

Keywords: BBP phase transition , Finite rank perturbations , Random matrix theory , spiked model , Stochastic Airy operator , Tracy–Widom distributions

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 4 • July 2016
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