The Continuum Directed Random Polymer

Speaker: 

Tom Alberts

Institution: 

Caltech

Time: 

Friday, March 16, 2012 - 1:00pm to 2:00pm

Location: 

Rowland 440R

The discrete directed polymer model is a well studied example of a Gibbsian disordered system and a random walk in a random environment. The usual goal is to understand how the random environment affects the behavior of the underlying walk and how this behavior varies with a temperature parameter that determines the strength of the environment. At infinite temperature the environment has no effect and the walk is the simple random walk, while at zero temperature the environment dominates and the walk follows a single path along which the environment is largest. For temperatures in between there is a competition between the walk wanting to behave diffusively (like simple random walk) and following a path of highest energy (like last passage percolation).

In this talk I will describe recent joint work with Kostya Khanin and Jeremy Quastel for taking a scaling limit of the directed polymer model to construct a continuous path in a continuum environment. We end up with a one-parameter family of random probability measures (indexed by the temperature parameter) that we call the continuum directed random polymer. As the temperature parameter varies the paths cross over from Brownian motion to what is conjectured to be a continuum limit of last passage percolation. This cross over is an inherent feature of the KPZ universality class, which I will also briefly describe.

Quenched asymptotics for Brownian motion in generalized Gaussian potential.

Speaker: 

Professor Xia Chen

Institution: 

University of Tennessee

Time: 

Tuesday, February 7, 2012 - 11:00am

Location: 

RH 306

Recall that the notion of
generalized function is introduced for the functions
that can not be defined pointwise, and
is given as a linear functional over the test functions.
The same idea applies to random fields. In this talk,
we study the quenched asymptotics for Brownian motion
in a generalized Gaussian field. The major ingredient
includes: Solution to
an open problem posted by Carmona and Molchanov (1995) with
an answer different from what was conjectured; the quenched
laws for Brownian motions in Newtonian-type potentials, and in the potentials
driven by white noise or by fractional white noise.

On diffusions interacting through their ranks

Speaker: 

Mikhaylo Shkolnikov

Institution: 

UC Berkeley

Time: 

Thursday, December 1, 2011 - 4:00pm

Location: 

RH 306

We will discuss systems of diffusion processes on the real line, in which the dynamics of every single process is determined by its rank in the entire particle system. Such systems arise in mathematical finance and statistical physics, and are related to heavy-traffic approximations of queueing networks. Motivated by the applications, we address questions about invariant distributions, convergence to equilibrium and concentration of measure for certain statistics, as well as hydrodynamic limits and large deviations for these particle systems. Parts of the talk are joint work with Amir Dembo, Tomoyuki Ichiba, Soumik Pal and Ofer Zeitouni

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