We study Gaussian sample covariance matrices with population covariance a bounded-rank perturbation of the identity, as well as Wigner matrices with bounded-rank additive perturbations. The top eigenvalues are known to exhibit a phase transition in the large size limit: with weak perturbations they follow Tracy-Widom statistics as in the unperturbed case, while above a threshold there are outliers with independent Gaussian fluctuations. Baik, Ben Arous and Péché (2005) described the transition in the complex case and conjectured a similar picture in the real case, the latter of most relevance to high-dimensional data analysis.
Resolving the conjecture, we prove that in all cases the top eigenvalues have a limit near the phase transition. Our starting point is the work of Rámirez, Rider and Virág (2006) on the general beta random matrix soft edge. For rank one perturbations, a modified tridiagonal form converges to the known random Schrödinger operator on the half-line but with a boundary condition that depends on the perturbation. For general finite-rank perturbations we develop a new band form; it converges to a limiting operator with matrix-valued potential. The low-lying eigenvalues describe the limit, jointly as the perturbation varies in a fixed subspace. Their laws are also characterized in terms of a diffusion related to Dyson's Brownian motion and in terms of a linear parabolic PDE.
We offer a related heuristic for the supercritical behaviour and rigorously treat the supercritical asymptotics of the ground state of the limiting operator.
In a further development, we use the PDE to make the first explicit connection between a general beta characterization and the celebrated Painlevé representations of Tracy and Widom (1993, 1996). In particular, for beta = 2,4 we give novel proofs of the latter.
Finally, we report briefly on evidence suggesting that the PDE provides a stable, even efficient method for numerical evaluation of the Tracy-Widom distributions, their general beta analogues and the deformations discussed and introduced here.
This thesis is based in part on work to be published jointly with Bálint Virág.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/31693 |
| Date | 05 January 2012 |
| Creators | Bloemendal, Alexander |
| Contributors | Virág, Bálint |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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