Limits of spiked random matrices II
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Alex Bloemendal [1] ; Bálint Virág [2]
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[1] Harvard University
Harvard University
City of Cambridge, Estados Unidos
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[2] University of Toronto
University of Toronto
Canadá
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- Localización: Annals of probability: An official journal of the Institute of Mathematical Statistics, ISSN 0091-1798, Vol. 44, Nº. 4, 2016, págs. 2726-2769
- Idioma: inglés
- DOI: 10.1214/15-AOP1033
- Enlaces
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Resumen
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.
