Random Fractals and Harmonic Measure

本项目主要研究调和测度的几何性质以及与调和测度相关的共形映射的性质。调和测度是复分析中的一个基本对象，它的多重分形结构为描述复平面上集合的共形几何提供了一个合适的框架。众所周知，调和测度在以分形集为界的域中表现出极值行为（即，非整数Hausdorff维数）。经验表明，在这种情况下，使用随机分形比使用确定性分形更有效。这样做的一个动机是，它允许人们在问题上使用广泛的强大的概率和分析工具。在物理学中，有许多随机分形作为晶格模型的尺度极限而出现，它们模拟了真实的物理现象。研究这类随机分形上调和测度的多重分形结构，可能有助于解决几何函数论中一些长期存在的问题--或者至少可以大大提高我们对它们的理解。这项研究还将揭示复分析和理论物理之间的联系。该项目与多重分形分析密切相关，多重分形分析是一个介于数学和物理之间的交叉学科。它强调复分析，概率论和统计物理之间的相互作用。所有这些领域都可以从这个项目中受益。例如，其中一些结果将使我们更深入地了解统计物理学的几个重要模型。此外，这一领域的某些成果与工程有着直接的联系。

This project focuses on the geometric properties of harmonic measure and properties of conformal mappings that are related to this classical conformal invariant. Harmonic measure is a fundamental object in complex analysis, and its multifractal structure provides an appropriate framework in which to describe the conformal geometry of sets in the complex plane. It is well known that harmonic measure exhibits its extremal behavior in domains that are bounded by fractal sets (i.e., sets of nonintegral Hausdorff dimension). Experience has shown that in this context it is more productive to work with random fractals than with deterministic ones. A motivating reason for doing so is that it allows one to bring to bear on problems a wide range of powerful probabilistic and analytic tools. There are a number of classes of random fractals that arise in physics as scaling limits of lattice models and that simulate real physical phenomena. Study of the multifractal structure of harmonic measure on such random fractals may help to solve some longstanding problems in geometric function theory -- or at least to improve significantly our understanding of them. The research will also shed new light on the linkages between complex analysis and theoretical physics. The project is closely related to multifractal analysis, an interdisciplinary subject that lies on the border between mathematics and physics. It emphasizes interaction between complex analysis, probability theory, and statistical physics. All these areas can benefit from this project. For example, some of the results will provide a deeper understanding of several important models from statistical physics. Moreover, certain results in this area have a direct connection to engineering.