Topics in Percolation & Particle Models

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This project comprises studies that seek to answer basic questions arising in two areas of Probability Theory, Interacting Particle Systems andPercolation. They include the construction and applicationof nucleation processestaking place at the touching points of forward and backwardpaths in the (double) Brownian web and the conceptually relatedmarking process on the ``double points'' of the (critical) continuumnonsimple loop process. This loop process, constructed by the PIand Federico Camia based on the Schramm-Loewner evolution (SLE)and the work of Schramm and Smirnov, is the full scaling limitfor two-dimensional critical percolation. The double points,where the loops touch themselves and each other, represent thescaling limits of lattice sites where a change in microscopic statehas a macroscopic effect on connectivity.There are many situations, such as the functioning of the Internetand 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 tonovel behavior of the entire system. A classic goal of ProbabilityTheory and Statistics is to understand these situations. A particularlyintriguing set of problems concerns so called ``critical'' systemswhere even one individual can provide a tipping point for the wholesystem. The goal of this research project is to lay a firm foundationfor the mathematics needed to understand such critical systems, at leastin some special situations where there have been theoretical breakthroughsin the past several years. It is hoped that progress in thesespecial cases will lead to more general progress.