Random Media in and Beyond Two Dimensions

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This project involves the study of mathematical models of real physical phenomena. Among these are so-called percolative models, which describe the flow of fluid in a porous medium, a growing bacterial infection, or a long polymer chain. Another model to be studied is the Bak-Tang-Wiesenfeld model, the classic model of self-organized criticality (SOC), where simple, local rules generate complex, large-scale patterns. SOC has been used to explain commonalities between such apparently dissimilar phenomena as solar flares and neuron firing. Many of the most successful approaches to problems in these areas are valid only in certain special cases; for instance, certain two-dimensional percolative models with particular symmetry properties. A main goal of the project is the development of robust mathematical techniques which explain the behavior of such systems -- for instance, how smooth is the surface of a spreading drop of fluid -- in a broad range of physically interesting cases.The specific research problems to be studied include the following. In critical Bernoulli percolation (where one randomly removes just enough edges to make an infinite graph finite), the PI will study scaling limits of graph components and give sharp asymptotics for graph distances and electrical resistances. In first-passage percolation (where one randomly perturbs edge lengths in a graph), the PI will study fluctuations of graph distances and the tortuosity of long geodesics. In the Bak-Tang-Wiesenfeld model (an interacting particle system), the PI will study the correlation between particle movements at different moments of time. Some related problems have previously seemed more tractable in special settings like certain "solvable" growth models or in certain dimensions. This project will employ robust methods – stochastic geometric techniques, concentration of measure methods, and others – which apply to the widest range of possible settings and which distill the general principles underlying phenomena of interest.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.