Random Fractals and Harmonic Measure

Princeton University, Princeton NJ

Investigators

Abstract

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.