LIMITS OF SPIKED RANDOM MATRICES II

Bloemendal, Alex and Virág, Bálint (2016) LIMITS OF SPIKED RANDOM MATRICES II. ANNALS OF PROBABILITY, 44 (4). pp. 2726-2769. ISSN 0091-1798

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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 Peche [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 Schrodinger operator on the half-line with r x 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 Painleve formulas for these r-parameter deformations of the GUE Tracy Widom law.

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