This project comprises studies that seek to answer basic questions arising in two areas of Probability Theory, Interacting Particle Systems and Percolation. They include the construction and application of nucleation processes taking place at the touching points of forward and backward paths in the (double) Brownian web and the conceptually related marking process on the ``double points'' of the (critical) continuum nonsimple loop process. This loop process, constructed by the PI and Federico Camia based on the Schramm-Loewner evolution (SLE) and the work of Schramm and Smirnov, is the full scaling limit for two-dimensional critical percolation. The double points, where the loops touch themselves and each other, represent the scaling limits of lattice sites where a change in microscopic state has a macroscopic effect on connectivity.
There are many situations, such as the functioning of the Internet and the behavior of engineered materials and of the economy, where many small individuals (web sites, molecules, investors) interact with each other in seemingly random ways leading to novel behavior of the entire system. A classic goal of Probability Theory and Statistics is to understand these situations. A particularly intriguing set of problems concerns so called ``critical'' systems where even one individual can provide a tipping point for the whole system. The goal of this research project is to lay a firm foundation for the mathematics needed to understand such critical systems, at least in some special situations where there have been theoretical breakthroughs in the past several years. It is hoped that progress in these special cases will lead to more general progress.
