The projects herein study spatial stochastic models, specifically percolation and short-range (realistic) spin glasses. The main questions concern first-passage percolation, independent percolation models at the critical threshold, and ground (zero-temperature) states of the Edwards-Anderson spin glass model. Viewed analytically, some of the projects seek to understand the structure and fluctuations of the maximum, or minimum, of a collection of correlated random variables; this applies not only to questions of geodesics and estimation of variance in first-passage percolation, but also to the organization of spin glass states. On the other hand the techniques have a geometric flavor, including tools from ergodic theory and Bernoulli percolation. It is expected that the results obtained will influence other mathematical areas, for instance polymer models, particle systems and questions in theoretical computer science.
The theory of random spatial systems has grown to include models of random distances, fluid flow through porous media, random growth models and magnetic properties of dilute metallic alloys. The projects studied here center on such models and concern the connection between the small-scale rules that define the random environment and the large-scale properties of the system. While some of these properties are highly dependent on the nature of the local rules, others are ``universal'' and seem to only depend on the dimension of the ambient space. A main goal of this project is to understand the origin of this universal behavior. The results should have consequences not only for related mathematical areas, but for topics of current interest in physics, computer science, biology and financial mathematics.
