Scaling limits of random curves at criticality

University Of Chicago, Chicago IL

Investigators

Abstract

Many mathematical models from statistical physics are characterized by the fact that there is a sharp qualitative change when a parameter reaches a certain value. This is an idealization of phase transitions such as the change of water from solid to liquid to gas as temperature varies. These mathematical models at critical values of the parameter often produce random fractals. The proposer will continue investigation of these fractals. The theory is well developed in two dimensions where the systems are invariant under conformal transformation and the investigation will be on fine properties. The theory in three dimensions is less developed and the main goal is to construct nontrivial models. More specifically, the goal in two dimensions is to study the Schramm-Loewner evolution (SLE) as a measure on curves parametrized with the natural fractal parametrization. Technical questions to study include the Holder continuity of the curves (under natural parametrization) and the stationary of the SLE loop measure. The latter property is needed to construct a conformally covariant measure on SLE loops analogous to the Brownian loop measure. There are several questions in three dimensions that will be investigated: trying to show that there is a unique chronological loop-erasure of Brownian motion and to describe the limit as a Laplacian random walk; proving analyticity of the intersection exponent for Brownian motion and establishing the multifractal spectrum for harmonic measure; and finding general methods to give nontrivial measures on continuous, non-self-intersecting curves of fractal dimension greater than one. The use of infinitesimal (non-standard) techniques will be considered as an alternative to continuum models for objects such as exponentials of Gaussian free fields, uniform spanning trees, and percolation.