Particle Systems, Percolation, and Scaling Limits

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In the general area of probability theory, an increasingly important role has been played in recent years 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 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. There are also connections to combinatorial optimization (as in the minimal spanning tree problem, which is a variant of the classical travelling salesman problem of designing optimal routings) and thus to various parts of theoretical computer science. This research project concerns the mathematical phenomena that occur in several representative examples of such stochastic systems with interesting spatial structure. One particular issue of study in the spanning tree problem is when an optimal route can go very far away from its starting and ending points -- a phenomenon known to experienced air travelers who want to optimize ticket price.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 specific topic is first-passage percolation based on d-dimensional Poisson point processes, with particular emphasis on nonexistence of doubly infinite geodesics. A second focus is on the classic problem of proving absence of an infinite cluster for intermediate dimensional critical Bernoulli percolation; a related problem concerns the number of trees in the minimal spanning forest and its dependence on spatial dimension. A third part of the project is to prove exponential decay of correlations in the recently constructed near-critical scaling limit of the two-dimensional Ising model. Finally, a fourth topic is the construction of scaling limits of voter model variants such as the q-state Potts model (in one space dimension) by using marked special space-time points of the Brownian web and net.