Large deviations from equilibrium measure for zeros of random holomorphic fields.

Speaker: 

Professor Steve Zelditch

Institution: 

Johns Hopkins

Time: 

Wednesday, April 8, 2009 - 2:00pm

Location: 

RH 306

An old result of Kac-Hammersley says that the complex zeros of a Gaussian
random polynomial \sum_{j = 0}^N a_j z^j with i.i.d. normal coefficients a_j, concentrate
on the unit circle. This seems counter intuitive at first, since the zeros could be anywhere.
We will explain this paradox and show that there is a very general result that empirical measures
of complex zeros tend to `equilibrium measures'. We then give a large deviations principle showing
that the probability of deviation from equilibrium measure is exponentially small.

Perturbed simple random walk

Speaker: 

Professor Ben Morris

Institution: 

UC Davis

Time: 

Tuesday, June 2, 2009 - 11:00am

Location: 

RH 306

A random walk is called recurrent if it is sure to return to its starting point and transient otherwise. A famous result of Polya is that simple symmetric random walk on the integer lattice is recurrent in dimensions 1 and 2, and transient in higher dimensions. We study random walks that are small perturbations of simple random walk. Our main result is that if the dimension is high enough then these random walks are transient.

Polymer Depinning Transitions with Loop Exponent One

Speaker: 

Professor Ken Alexander

Institution: 

USC

Time: 

Tuesday, April 28, 2009 - 11:00am

Location: 

RH 306

We consider a polymer with configuration modeled by the trajectory of a Markov chain, interacting with a potential of form u+V_n when it visits a particular state 0 at time n, with V_n representing i.i.d. quenched disorder. There is a critical value of u above which the polymer is pinned by the potential. Typically the probability of an excursion of length n for the underlying Markov chain is taken to decay as a power of n (called the loop exponent), perhaps with a slowly varying correction. A particular case not covered in a number of previous studies is that of loop exponent one, which includes simple random walk in two dimensions. We show that in this case, at all temperatures, the critical values of u in the quenched and annealed models are equal, in contrast to all other loop exponents, for which these critical values are known to differ at least at low temperatures. The work is joint with N. Zygouras.

"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.)

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