Limits of spiked random matrices II

Details Limits of spiked random matrices II Limits of spiked random matrices II

2011-09-16

arxiv.org/abs/1109.3704v1

Mathematics

Probability

46 pages

Scientific paper

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 P\'ech\'e (2005). 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 Schr\"odinger 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 over all perturbations in a fixed subspace. We treat the real, complex and quaternion (beta = 1,2,4) cases simultaneously. We also characterize the limit laws in terms of a diffusion related to Dyson's Brownian motion, and further in terms of a linear parabolic PDE; here beta is simply a parameter. At beta = 2 the PDE appears to reconcile with known Painlev\'e formulas for these r-parameter deformations of the GUE Tracy-Widom law.

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