There has been significant progress in the last few years in the rigorous understanding of two-dimensional lattice models in statistical physics "at criticality". A number of continuous models, most particularly the Schramm-Loewner evolution (SLE), have been constructed, and in some cases it has been proved that discrete models approach SLE in the limit. The proposer will study a number of discrete models, e.g., self-avoiding walk, Laplacian random walks, and walks on certain random graphs, with the hope of showing that they also converge to SLE. Also, the proposer will consider models in three dimensions where the recently developed techniques which rely on conformal invariance to not apply. The goal is to find three-dimensional continuous models to be candidates for limits of discrete systems. The goal of this proposal is to construct and analyze mathematical models for phase transition, which is the study of the sharp changes in a physical system when changing a parameter such as the freezing of water when the temperature is reduced. More generally, the mathematical goal is to understand universality principles that allow one to predict macroscopic behavior from microscopic rules. As well as being important to probability theory, the results will be relevant to many areas of theoretical physics. Special focus will be placed on the approach to the limit for two-dimensional models where the limit itself is now well understood and to construct candidates for the limit in three dimensions where the problems are more challenging.
