The Annals of Probability
- Ann. Probab.
- Volume 44, Number 4 (2016), 2726-2769.
Limits of spiked random matrices II
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.
Ann. Probab., Volume 44, Number 4 (2016), 2726-2769.
Received: June 2012
Revised: May 2015
First available in Project Euclid: 2 August 2016
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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