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.
This project in the mathematical area of Probability Theory is concerned with the large scale geometric effects of randomness that initially occurs only on small scales. Although the mathematical models studied are relatively simple to define, the rigorous analysis of their properties requires the development of new mathematical techniques. As a parameter of the model is varied, there is a critical value when the large scale geometric properties change. One is interested primarily in the behavior at or near this critical value. More complicated models of this general sort have been used to describe such diverse phenomena as the reflective properties of Antarctic sea ice when the temperature or salinity varies and the gridlock effects of increasing density of cars in urban transportation. One classic model, known as percolation, can be thought of as a transportation model in which a fraction F of small scale passageways are blocked. When F exceeds a critical value all (large scale) motion is blocked. A classic problem is to prove that practically all motion is already blocked as the critical value is approached. One of the outcomes of this project was to extend the previously known result that this is the case in a strictly two-dimensional setting into the third dimension (e.g., a city with two levels of roadways). Motivated by that outcome, more progress into three dimensions has been obtained since. One impact of this project on human resources was the training of two US PhD students who collaborated on some of the research supported by this grant.
