"Continuum limits for beta ensembles"

Speaker: 

Professor Brian Rider

Institution: 

University of Colorado

Time: 

Tuesday, March 3, 2009 - 11:00am

Location: 

RH 306

The beta ensembles of random matrix theory are natural generalizations of the Gaussian Orthogonal, Unitary, and Symplectic Ensembles, these classical cases corresponding to beta = 1, 2, and 4. We prove that the extremal eigenvalues for the general ensembles have limit laws described by the low lying spectrum of certain raandom Schroedinger operators, as conjectured by Edelman-Sutton. As a corollary, a second characterization of these laws is made the explosion probability of a simple one-dimensional diffusion. A complementary pictures is developed for beta versions of random sample-covariance matrices. (Based on work with J. Ramirez and B. Virag.)

On the spectrum of large random reversible stochastic matrice

Speaker: 

Professor Pietro Caputo

Institution: 

Universit Roma Tre

Time: 

Tuesday, February 3, 2009 - 11:00am

Location: 

RH 306

We consider random matrices associated to random walks on the complete
graph with random weights. When the weights have finite second moment we
find Wigner-like behavior for the empirical spectral density. If the
weights have finite fourth moment we prove convergence of extremal
eigenvalues to the edge of the semi-circle law. The case of weights with
infinite second moment is also considered. In this case we prove
convergence of the spectral density on a suitable scale and the limiting
measure is characterized in terms certain Poisson weighted infinite
trees associated to the starting graph. Connections with recent work on
random matrices with i.i.d. heavy-tailed entries and several open
problems are also discussed. This is recent work in collaboration with
D. Chafai and C. Bordenave (from Univ. P.Sabatier, Toulouse - France).

Heat kernel estimates for Dirichlet fractional Laplacian

Speaker: 

Professor Panki Kim

Institution: 

Seoul National University

Time: 

Tuesday, January 20, 2009 - 11:00am

Location: 

RH 306

In this talk, we discuss the sharp two-sided estimates for the heat kernel of Dirichlet fractional Laplacian in open sets. This heat kernel is also the transition density of a rotationally symmetric -stable process killed upon leaving an open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets. This is a joint work with Zhen-Qing Chen and Renming Song.

On the almost-sure invariance principle for random walk in random environment

Speaker: 

Professor Eiras Rassoul-Agha

Institution: 

University of Utah

Time: 

Tuesday, January 13, 2009 - 11:00am

Location: 

RH 306

Consider a crystal formed of two types of atoms placed at the nodes of the
integer lattice. The type of each atom is chosen at random, but the crystal
is statistically shift-invariant. Consider next an electron hopping from atom
to atom. This electron performs a random walk on the integer lattice with
randomly chosen transition probabilities (since the configuration seen by
the electron is different at each lattice site). This process is highly
non-Markovian, due to the interaction between the walk and the
environment.

We will present a martingale approach to proving the invariance principle
(i.e. Gaussian fluctuations from the mean) for (irreversible) Markov chains
and show how this can be transferred to a result for the above process
(called random walk in random environment).

This is joint work with Timo Sepp\"al\"ainen.

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