Self-Avoiding Models: Polymers and Surfaces (Carl Mueller -- University of Rochester)

This is joint work with Eyal Neuman. Polymer models give rise to some of the most challenging problems in probability and statistical physics. We typically model a polymer using a random walk, where the time parameter $n$ of the walk represents distance along the polymer starting from one end. That is, we imagine that the polymer is built up by adding molecules one by one at random angles. We usually include a self-avoidance term, reflecting the idea that different parts of the polymer cannot be in the same place at the same time. A difficult problem, unsolved in the most important physical cases, is to predict the end-to-end distance or radius of the polymer. In this talk, I will discuss two extensions of the random polymer model. (1) Moving polymers can be modeled by stochastic partial differential equations. If the polymer takes values in one-dimensional Euclidean space, we give fairly sharp upper and lower bounds for its radius. We find that there is more stretching than in typical one-dimensional polymer models that do not have time dependence. (2) Random surfaces can be modeled by elastic manifolds, also called discrete Gaussian free fields. Free fields originate in quantum field theory. If the dimensions of the parameter space and the range are the same, we can derive bounds on the radius of the polymer. These bounds are fairly sharp in two dimensions. We will explain the models mentioned above and give an outline of our proof techniques.