Scaling limits of random and deterministic diffusion processes on the integer lattice

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The principal investigator will study mathematical models of microscopic interactions in solid materials. More specifically, the PI will study the scaling limits of such models, which capture macroscopic effects of microscopic structure. The PI is interested in three important classes of models arising from statistical physics. First, there are cluster growth models for corrosion of metals and crystal formation. Second, there are models of electrical properties of randomly composited materials. Third, there are models which lie in the intersection of the previous two categories, such as those for fluid flow in porous media. Many unresolved questions surround these models and are now of great interest in probability theory. The PI aims to resolve some of these questions by (1) using computer simulation to identify new phenomena and (2) applying tools from analysis, combinatorics, and probability to rigorously understand the new phenomena.The models alluded to above share a common theme; each is a diffusion processes on the integer lattice whose scaling limit can be interpreted as the solution of a nonlinear elliptic partial differential equation (PDE). The PI is primarily interested in three models: the Abelian sandpile, random walks in random environments, and first passage percolation. The PI will study the interplay between the asymptotic statistics of the diffusion process and the regularity properties of the solutions of the limiting PDE. Since the two sides here are tightly coupled, it is often possible and fruitful to transfer results between them. The strategy will be to carefully adapt the regularity machinery of the limiting PDE to the discrete setting, since this occasionally leads to the discovery of new phenomena.