Studies in First-Passage Percolation and Random Walks with Scenery

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9815226 Howard The work under this grant comprises two areas of investigation in the general category of probability theory. The first concerns rates of convergence (to an associated asymptotic shape) and related properties of infinite geodesics in the context of certain Euclidean first-passage percolation models (Euclidean FPP). Rigorous unconditional results about infinite geodesics (time-minimizing paths) in the context of standard FPP are quite sparse, largely due to technical difficulties associated with lattice effects. Models of Euclidean FPP take place on graphs constructed from a homogeneous Poisson process and their statistical properties enjoy complete invariance under all rigid motions. In this sense, Euclidean models are a natural setting in which to study FPP geodesics. The second area concerns a host of problems arising from random walks on the integers where some fixed but possibly random coloring, or "scenery", is assigned to the integers. In this context, a random walk on the integers produces a record of scenery "observed" along the walk. The issues under investigation concern, loosely speaking, what can be inferred about the scenery by observing one such scenery record. The difficulty stems from the fact that one does not know the steps taken by the walk, only the scenery observed along the walk. Both areas of investigation involve stochastic processes associated with some type of spatial structure. While the focus of this research is the mathematical aspects of these models, models of this general sort arise quite naturally in connection with a number of interesting physical phenomena. For example, the standard FPP model was first introduced as a representation of fluid flow through a random porous medium such as an aquifer or, on a smaller physical scale, a membrane. There are also connections to other aspects of materials science, including the formation and propagation of cracks.