Topics in Percolation and Particle Models

New York University, New York NY

Investigators

Abstract

The work under this grant is in the general area of Probability Theory with special emphasis on a number of stochastic models with interesting spatial structure. One project, with C.D. Howard, concerns Euclidean first-passage percolation and aims to prove the nonexistence of doubly infinite geodesics and derive the values of two-dimensional fluctuation exponents that to date have only been proved using exact solution methods for models with special combinatorial structure. Other collaborative projects are on interacting particle systems. One is with L.R. Fontes, M. Isopi and K. Ravishankar and concerns aging, scaling limits and chaotic time dependence in such systems as voter models with random rates. Another is with F. Camia, E. De Santis and others and concerns local transience, recurrence and absorption issues for zero-temperature stochastic Ising models, including those with random environments. In the general area of probability theory, an increasingly important role is played by systems in which random effects are observed in the spatial structure rather than in, or in addition to, the behavior as a function of time (as in models of equity prices). Some of these models, such as first-passage percolation, have arisen separately in multiple contexts, such as fluid flow in porous media (which is relevant for example to modelling of pollutant dispersion in aquifers), polymer structure and other parts of materials science. The research under this grant concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. Although the main focus is on rigorous mathematical results for simplified models, the long term goal is to provide new understanding and tools that will be useful for the more complex models used in applications.