Critical and near critical systems in statistical mechanics

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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.