Critical planar systems and conformal invariance

Columbia University, New York NY

Investigators

Abstract

The last decade has seen a series of major achievements in the study of conformally invariant systems, in particular following the introduction of Schramm-Loewner Evolutions. This new approach has proved extremely effective in describing the scaling limit of interfaces of critical models of two-dimensional Statistical Physics, such as percolation and the Ising model, in accordance with the earlier predictions of Conformal Field Theory (CFT). The research program proposed here focuses on systems on paths and loops, in particular in relation with the CFT formalism; on free field invariance principles and the analysis of Cauchy-Riemann operators; and on the connections between geometric and functional representations of continuous scaling limits. The goal of this proposal is to analyze mathematical models of the physical phenomenon of phase transition. A phase transition describes a sharp qualitative change in a physical system under variation of an external parameter, such as freezing of water (transition from liquid to solid phase as temperature decreases). At the phase transition, a random macroscopic geometry may emerge, akin to the wiggly interfaces separating non mixing fluids like oil and water, or positively and negatively charged regions in a magnetic material. The study of those fluctuating interfaces is grounded in both Probability Theory and Theoretical Physics. Special emphasis will be put on the interplay between these two approaches, and their relation to other areas of mathematics.