Resumen de Limits of spiked random matrices II
Alex Bloemendal, Bálint Virág
The top eigenvalues of rank rr 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)(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×rr×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 (β=1,2,4β=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 ββ appears simply as a parameter. At β=2β=2, the PDE appears to reconcile with known Painlevé formulas for these rr-parameter deformations of the GUE Tracy–Widom law.
