Limits of spiked random matrices II

TL;DR

This paper characterizes the limiting distributions of top eigenvalues in spiked random matrices near critical perturbations, resolving a longstanding conjecture and connecting the results to diffusion processes and PDEs.

Contribution

It introduces a new algebraic $(2r+1)$-diagonal form and links the eigenvalue limits to a matrix-valued Schrödinger operator, diffusion processes, and PDEs, fully resolving the conjecture.

Findings

Derived limiting distributions for near-critical perturbations.

Unified treatment of real, complex, and quaternion cases.

Connected eigenvalue limits to diffusion processes and Painlevé formulas.

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 P\'{e}ch\'{e} [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 Schr\"{o}dinger 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 Painlev\'{e} formulas for these $r$-parameter deformations of the GUE Tracy-Widom law.