Discrete models and conformally invariant limits

Columbia University, New York NY

Investigators

Abstract

In statistical mechanics and probability theory, systems consisting of a large number of microscopic elements - such as molecules - individually subject to randomness or noise, and interacting with each other, play a prominent role. It is a fundamental problem to understand how simple, microscopic interaction rules produce intricate random features on a macroscopic scale, features that can be modeled by continuous stochastic structures. This general area of problems is relevant to many important physical phenomena, such as porosity and ferromagnetism. The project's emphasis is on the case of two-dimensional systems, where the last fifteen years have seen spectacular and ongoing progress, involving a wide variety of mathematical concepts and techniques, along with a rapprochement between mathematical and theoretical physics. The PI will continue to engage in mathematical and interdisciplinary training activities of students and junior researchers, in particular in relation with his research projects at the interface of Physics and Mathematics. The program of research described in this project is mainly focused on two-dimensional critical models in statistical physics, with special emphasis on conformal invariance and the interplay between the field formulation and the geometric aspects of their scaling limits. The introduction of Schramm-Loewner evolutions (SLE) has deeply influenced the study of conformally invariant random systems. This new class of stochastic continuous planar curves has proved extremely effective in describing the scaling limit of interfaces of critical models of statistical physics, such as percolation and the Ising model. The research projects of the PI have three main focuses. The first is the study of systems of SLE-type paths and loops, in particular in relation with the Conformal Field Theory formalism. The second concerns the dimer model and aspects of the free field description of its scaling limit, building on the analysis of families of Cauchy-Riemann operators. The third focus involves the exact relations between SLE paths and an ambient Gaussian field. The expectation of the PI is that these studies will help provide a deeper understanding of critical models and the interplay of the relevant concepts and tools in combinatorics, probability, analysis, complex geometry and representation theory.