We consider the Williams Bjerknes model, also known as the biased voter model on the d-regular tree T^d, where d \geq 3. Starting from an initial configuration of ``healthy'' and ``infected'' vertices, infected vertices infect their neighbors at Poisson rate \lambda \geq 1, while healthy vertices heal their neighbors at Poisson rate 1. All vertices act independently. It is well known that starting from a configuration with a positive but finite number of infected vertices, infected vertices will continue to exist at all time with positive probability iff \lambda > 1. We show that there exists a threshold \lambda_c \in (1, \infty) such that if \lambda > \lambda_c then in the above setting with positive probability all vertices will become eventually infected forever, while if \lambda < \lambda_c, all vertices will become eventually healthy with probability 1. In particular, this yields a complete convergence theorem for the model and its dual, a certain branching coalescing random walk on T^d -- above \lambda_c. We also treat the case of initial configurations chosen according to a distribution which is invariant or ergodic with respect to the group of automorphisms of T^d. Joint work with A. Vandenberg-Rodes, R. Tessler.

The aim of this talk is to introduce the subject of spin glasses,
and more generally the statistical mechanics of quenched disorder,
as a problem of general interest to physicists from multiple disciplines and
backgrounds. Despite years of study, the physics of quenched
disorder remains poorly understood, and represents a major gap in our
understanding of the condensed state of matter. While there are many
active areas of investigation in this field, I will narrow the focus of this
talk to our current level of understanding of the low-temperature
equilibrium structure of
realistic (i.e., finite-dimensional) spin glasses.

I will begin with a brief survey of why the subject is of interest not only
to physicists,
but also mathematicians, computer scientists, and scientists working in
other areas. A brief review of the basic features of spin glasses and what
is
known experimentally will follow. I will then turn to the problem of
understanding the nature of the spin glass phase --- if it exists.
The central question to be addressed is the nature of broken symmetry in
these systems. Parisi's replica symmetry breaking approach,
now mostly verified for mean field spin glasses, attracted great excitement
and interest as a novel and exotic form of symmetry breaking. But does it
hold also for real spin glasses in finite dimensions? This has been a
subject of intense controversy, and although the issues surrounding it have
become more sharply defined
in recent years, it remains an open question. I will explore this problem,
introducing new mathematical constructs such as the metastate along the way.
The talk will conclude with an examination of how and in which respects the
statistical mechanics of disordered systems might differ from that of
homogeneous systems.

Abstract: For the site percolation model on the triangular lattice and
certain generalizations for which Cardy’s Formula has been established
we acquire a power law estimate for the rate of convergence of the
crossing probabilities to Cardy’s Formula.

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.

We derive a type of KPZ equation as a scaling limit of fluctuation fields in weakly asymmetric particle systems such as simple exclusion and zero-range processes.  Joint work (in progress) with Milton Jara and Patricia Goncalves.