Schramm-Loewner Evolution and Other Scaling Limits

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The Schramm-Loewner evolution (SLE) is a continuous model of two-dimensional systems in statistical mechanics at criticality.The proposer will study the detailed fractal and multifractal properties of SLE paths. A goal is the further understanding of the relationship between microscopic rules and macroscopic behavior for critical phenomena and the effect of boundary conditions and other global geometry on the behavior. Another is to establish the multifractal formalism for a model with nontrivial self-repulsion interaction. A long-range hope is to use this structure to understand configurational measures on discrete paths such as the problem of the self-avoiding random walk.The proposer will also try to understand what ideas can extend to dimensions other than two, in particular for random walks with self-repulsions in three dimensions, where conformal invariance is not expected. In higher dimensions, the loop-erased walk, Laplacian walk with exponent and continuous analogues, and Brownian intersection problems will be studied.The study of critical phenomenon, i.e. the behavior of a system at or near the point at which it changes state, leads to a number of mathematical constructions. For example, interfaces between different phases or materials can be viewed as a curve or a surface. At criticality, these curves and surfaces have ``fractal'' behavior which means that they have scaling properties like spaces of unusual, often fractional, dimension. For two dimensional systems (or three dimensional systems constrained so that they are almost two dimensional), a stronger property called conformal invariance has been observed. The proposer will continue study of a major new model in this area, the Schramm-Loewner evolution (SLE) with a particular emphasis on the detailed fractal geometry of the curve and the interaction of the curve with outside boundaries or walls. The proposer will also explore similar questions in three dimensions which are of great interest, but much more difficult because of the lack of conformal invariance as a tool.