Limits of spiked random matrices II

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 Baik, Ben Arous and Péché [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 Schrödinger operator on the half-line with r × 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, 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 β appears simply as a parameter. At β = 2, the PDE appears to reconcile with known Painlevé formulas for these r-parameter deformations of the GUE Tracy-Widom law.

Spiking the random matrix hard edge

“…We show that if nΣ r has an appropriate limit then there is a new family of random operator limits indexed by these spiking parameters. These spiked hard edge operators produce diffusion/PDE descriptions for the new limit laws (in the manner familiar from [3,4,18,19]), and, again as expected, recapture all known spiked Tracy-Widom laws in the a → ∞ limit.…”

“…The results of [3,4] follow upon those of [19] in that they hinge on tridiagonal representations for the spiked Wishart ensembles. For r = 1 these are the same random tridiagonal matrices introduced by Dumitriu-Edelman [10], which subsequently formed the basis for the conjectured Stochastic Airy Operator formulation of the Tracy-Widom(β) laws in [11,26] (and again proved in [19]).…”

“…In another direction, Bloemendal-Virág [3,4] proved that the β = 1, 2, or 4 soft-edge (critically) spiked laws can be characterized in a unified way through the eigenvalue problem for the random operator H acting on functions f ∈ L 2 [[0, ∞), F r ] defined by…”

“…That is, B σ B T σ has the same eigenvalues as XΣX † in the real/complex/quaternion (β = 1/2/4) cases, and afterwards this ensemble may be declared the general one-spiked model. For r > 1, one of the key contributions of [4] is a block tridiagonalization procedure (which we revisit in Section 3) that commutes with r-fold spiking. Unfortunately, this model does not immediately generalize to all β > 0.…”

“…We also show that, under a natural limit, all spiked hard edge laws derived here degenerate to the critically spiked soft edge laws (or deformed Tracy-Widom laws). The latter were first described at β = 2 by Baik, Ben Arous, and Peché [2], and from a unified β random operator point of view by Bloemendal and Virág [3,4]. …”