Random conformal snowflakes


In many problems of classical analysis extremal configurations appear to exhibit complicated fractal structures, making it hard to describe them and to attack such problems. This is particularly true for questions related to the multifractal analysis of harmonic measure. We argue that, searching for extremals in such problems, one should work with random fractals rather than deterministic ones. We introduce a new class of fractals: random conformal snowflakes, and investigate their properties, developing tools to estimate spectra and showing that extremals can be found in this class. As an application we significantly improve known estimates from below on the extremal behavior of harmonic measure, showing how to construct a rather simple snowflake, which has a spectrum quite close to the conjectured extremal value.


Dmitri Beliaev

Department of Mathematics, Princeton University, Fine Hall - Washington Rd., Princeton, NJ 08544, United States

Stanislav Smirnov

Université de Genève, Section de Mathématiques, 2-4, rue du Lièvre, c.p. 64, CH-1211 Genève 4, Switzerland