Near-critical two-dimensional random systems

New York University, New York NY

Investigators

Abstract

This project studies random spatial systems, such as percolation or the Ising model of ferromagnetism. These models all exhibit the statistical physics phenomenon of a phase transition, their macroscopic behavior changing drastically as one parameter varies: there is a sharp transition for a particular critical value of this parameter. At the critical point exactly, random macroscopic - most often fractal - geometries arise, and in two dimensions these geometries are widely believed to possess a strong property of conformal invariance. The mathematical understanding of these models has considerably improved over the last ten years following breakthroughs of Lawler, Schramm, Smirnov and Werner, in particular thanks to the conformally invariant Schramm-Loewner-Evolution process of stochastic growth in the plane. This project addresses questions related to the behavior of these models at and near criticality, as well as related "self-critical" systems - forest-fire models for instance - where a phase transition intrisically appears without any fine-tuning of a parameter. The area of probability theory studied here overlaps combinatorics and complex analysis, and it also uses ideas and techniques from statistical mechanics. Random shapes, such as rough interfaces created by welding two metals, or irregular sea coasts fashioned by erosion, are omnipresent in nature. These shapes usually display a fractal behavior, and an increasingly important part of probability theory is devoted to studying such models where spatial randomness plays a central role. This leads to deep and fascinating mathematical questions, in particular surprising "universality" properties arise: for instance, similar shapes appear in situations that are, at first sight, completely unrelated, based on totally different physical, chemical or biological mechanisms. A better mathematical understanding of simplified models would provide new insight on more complex models used in applications.