Schramm-Loewner Evolution and Other Scaling Limits

Schramm-Loewner演化（SLE）是统计力学中二维系统在临界状态下的连续模型。本文将详细研究SLE路径的分形和多重分形特性。目标是进一步理解临界现象的微观规则和宏观行为之间的关系，以及边界条件和其他全局几何对行为的影响。另一个是建立具有非平凡自斥相互作用模型的多重分形形式。一个长期的希望是使用这种结构来理解离散路径上的构型度量，比如自我避免随机游走的问题。提议者还将尝试理解哪些想法可以扩展到二维以外的维度，特别是在三维中具有自排斥的随机行走，其中不期望共形不变性。在高维中，将研究环擦除行走、带指数和连续类似物的拉普拉斯行走以及布朗交叉问题。对临界现象的研究，即系统在其改变状态的点或点附近的行为，导致了许多数学结构。例如，不同相或材料之间的界面可以看作是一条曲线或一个表面。在临界状态下，这些曲线和曲面具有“分形”行为，这意味着它们具有缩放特性，就像不寻常的、通常是分数维的空间一样。对于二维系统（或三维系统的约束，使他们几乎是二维的），已经观察到一个更强的性质，称为共形不变性。申请人将继续研究该领域的一个主要新模型，Schramm-Loewner演化（SLE），特别强调曲线的详细分形几何以及曲线与外部边界或墙壁的相互作用。作者还将在三维空间中探索类似的问题，这些问题非常有趣，但由于缺乏共形不变性作为工具而更加困难。

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.