Particle Systems and Scaling Limits in Two (and More) Dimensions

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This project comprises studies that seek to answer basic questions arising in interacting particle systems, including percolation and Ising models. They include the construction and application of natural measures as limits of rescaled counting measures of lattice points with special macroscopic properties. For coalescing one-dimensional random walks and their Brownian web scaling limit in two-dimensional space-time, the special points are type (0,2) where zero paths enter and two leave and type (1,2) where one path enters and two leave. In d-dimensional Ising-related random cluster percolation models, the special points are where k disjoint paths emerge, with a focus onk=1 and d=2 to obtain limit cluster area measures and Euclidean random fields in terms of measure ensembles.It is also proposed to attack open problems concerning coarsening dynamics for d equal two or more, invasion percolation for d equal three or more and d=3 Ising models.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. A major goal of this research project is to lay a firm foundation for the mathematics needed to understand such critical systems.It is hoped that progress for certain special cases will lead to more general approaches.