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.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 (.=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 . appears simply as a parameter. At .=2, the PDE appears to reconcile with known Painlevé formulas for these r-parameter deformations of the GUE Tracy.Widom law.