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Limits of spiked random matrices. II. (English) Zbl 1396.60004
Author’s abstract: 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 J. Baik [Duke Math. J. 133, No. 2, 205–235 (2006; Zbl 1139.33006)]. 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 $$r\times r$$ 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 Painlevé formulas for these $$r$$-parameter deformations of the GUE Tracy-Widom law.
For Part I, see [the authors, Probab. Theory Relat. Fields 156, No. 3–4, 795–825 (2013; Zbl 1356.60014)].

##### MSC:
 60B20 Random matrices (probabilistic aspects) 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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