Schramm-Loewner Evolution and Other Scaling Limits

Schramm-Loewner 演化和其他缩放限制

• 批准号: 0907143
• 负责人: Gregory Lawler
• 金额: $ 70万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-15 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0907143&HistoricalAwards=false
• 关键词: Schramm; Loewner; Evolution; Other; Scaling

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.

Schramm-Loewner演化(SLE)是统计力学中二维系统在临界点的连续模型。目标是进一步了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。二是建立了具有非平凡自排斥相互作用的模型的多重分形形式。一个长远的希望是利用这种结构来理解离散路径上的构形测量，比如自我避免随机行走的问题。提出者还将试图理解哪些想法可以扩展到二维以外的其他维度，特别是对于三维空间中具有自排斥的随机行走，其中共形不变性是不被期望的。在高维中，我们将研究环擦除行走、具有指数和连续相似的拉普拉斯行走以及布朗相交问题。对临界现象的研究，即系统在其改变状态的点或附近的行为，将导致许多数学构造。例如，可以将不同相或材质之间的界面视为曲线或曲面。在临界点，这些曲线和曲面具有“分形性”行为，这意味着它们具有不寻常的、通常是分数维的空间的缩放特性。对于二维系统(或被约束为几乎是二维的三维系统)，观察到了一种更强的称为共形不变性的性质。作者将继续研究这一领域的一个重要的新模型--Schramm-Loewner演化(SLE)，特别强调曲线的详细的分形几何以及曲线与外部边界或墙的相互作用。提出者还将在三维空间中探索类似的问题，这些问题很有兴趣，但由于缺乏共形不变性作为工具，因此难度要大得多。