LIMITS OF SPIKED RANDOM MATRICES II

• 成果类型: Article
• 署名作者: Bloemendal, Alex; Virag, Balint
• 署名单位: Harvard University; University of Toronto
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/15-AOP1033
• 发表日期: 2016
• 页码: 2726-2769
• 关键词: largest eigenvalue; distributions

摘要

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.