Continuum limits of spiked random matrices

Friday, June 22, 2012 - 2:00pm - 3:15pm
Lind 305
Alex Bloemendal (Harvard University)
I will discuss the effect of finite rank perturbations (or spikes) on the random matrix soft edge, focusing on the phase transition discovered in the complex setting by Baik, Ben Arous and Péché. While small spikes leave the usual Tracy-Widom fluctuations of the top eigenvalues unaltered, large spikes lead to outliers with Gaussian fluctuations. New structure emerges around the transition point with near-critical spikes deforming the soft edge limit. Understanding this transition regime in the real case remained open for some time.

I will describe joint work with B. Virág that treats the phase transition in a general beta setting. With a single spike one obtains the usual limiting random Schrödinger operator on the half-line but with a modified boundary condition depending on the spike; to deal with several spikes we develop a matrix-valued analogue. The resulting deformations of the Tracy-Widom laws can be further characterized in terms of a diffusion (related to Dyson's Brownian motion in the higher-rank case) or a linear parabolic PDE. The latter can be connected with the known Painlevé II structure and, separately, seems effective for numerical evaluation.