This project will study systems from statistical mechanics and probability that exhibit critical phenomena. The systems include self-avoiding random walks and loops, Ising spin systems, and percolation. Our first approach is through real space renormalization group transformations. These transformations for Ising type models relate the critical behavior of the model to the behavior of the renormalization group map near the critical point. This map is not well defined mathematically, and the project will study the mathematical properties of this map. Our second approach is restricted to two dimensional systems where conformal invariance miraculously appears when there is critical behavior. The research will study the relation between the Schramm-Loewner evolution and various discrete models by studying the Loewner driving process, especially for models which are near but not at their critical point. The relation of the bi-infinite self-avoiding walk to the newly discovered conformally invariant measures on self-avoiding loops in the plane will also be investigated.
The systems that will be studied contain randomness at a microscopic length scale. For most values of the parameters, e.g., temperature, this randomness is not seen at the macroscopic scale. But for special values of the parameters this microscopic randomness can produce macroscopic effects. This is one characterization of a phase transition. Physicists have developed powerful techniques for studying phase transitions, but the mathematics of these methods is not well understood. This research will further our understanding of critical phenomena by developing the mathematics behind two of the most important of the physicist's tools - the renormalization group and conformal invariance. The research on the renormalization group will focus on the Ising model, arguably the single most important model of the magnetic behavior of crystalline materials. One product of the research on conformal invariance will be a much faster algorithm for numerically computing conformal maps which are used extensively in science and engineering.
This project studied systems from statistical mechanics and probability that exhibit critical phenomena. The two most important systems studied were the self-avoiding walk and the Ising model. The self-avoiding walk is a model of a random walk that is not allowed to visit a location more than once. The Ising model is a model for the spins in a crystal which can be in an up or down state. At low temperatures the model favors adjacent spins pointing in the same direction, while at high temperature the spins are "thermal idiots" which do not care what their neighbors are doing. These models contain randomness at a microscopic length scale. For most values of the parameters, e.g., temperature, this randomness is not seen at the macroscopic scale. But for special values of the parameters this microscopic randomness can produce macroscopic effects. This is one characterization of a phase transition. Physicists have developed powerful techniques for studying phase transitions, but the mathematics of these methods is not well understood. This research contributed to our understanding of critical phenomena by developing the mathematics behind two of the most important of the physicist's tools - the renormalization group and conformal invariance. Our research on renormalization group transformations focused on a particular transformation know as majority rule. This transformation has been widely used in numerical studies of the renormalization group and appeared to work quite well. The conventional wisdom is that this transformation may be well approximated by a finite dimensional transformation with relatively small dimension. Our results cast grave doubt on this conventional wisdom and raise the interesting question of just why these renormalization group maps work at all. One of the most important quantities to understand for the self avoiding walk is how it exits a bounded domain, i.e., how the exit point is distributed over the boundary. Our research on the conformal invariance of the self-avoiding walk has furthered our understanding of this exit distribution and the boundary effects that must be carefully taken into account to understand it. Our research also further our understanding of how these self-avoiding walks can be written in terms of simple units and how these walks may be simulated more efficiently. Four graduate students were extensively involved in this research. A group consisting of the PI, these graduate students and several undergraduates met on a weekly basis to work on some of the topics in this research.
