## Departmental PhD Thesis Exam – Alexander Bloemendal

Everyone welcome. Refreshments will be served in the Math Lounge
before the exam.

DEPARTMENTAL PHD THESIS EXAM
----------------------------

Wednesday, July 27, 2011
3:10 p.m., in BA 6183, 40 St. George Street

PhD Candidate:  Alexander Bloemendal

Thesis Title:  Finite Rank Perturbations of Random Matrices and Their
Continuum Limits
(http://www.math.toronto.edu/alexb/Bloemendal_thesis.pdf)

Thesis Abstract:
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 motionand 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.