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Limiting Shape for Directed Percolation Models

James B. Martin
The Annals of Probability
Vol. 32, No. 4 (Oct., 2004), pp. 2908-2937
Stable URL: http://www.jstor.org/stable/3481510
Page Count: 30
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Limiting Shape for Directed Percolation Models
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Abstract

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z+ d, d ≥ 2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits $g({\bf x})={\rm lim}_{n\rightarrow \infty }n^{-1}T(\lfloor n{\bf x}\rfloor)$ exist and are constant a.s. for x∈ R+ d, where T(z) is the passage time from the origin to the vertex z∈ Z+ d. We show that this shape function g is continuous on R+ d, in particular at the boundaries. In two dimensions, we give more precise asymptotics for the behavior of g near the boundaries; these asymptotics depend on the common weight distribution only through its mean and variance. In addition we discuss growth models which are naturally associated to the percolation processes, giving a shape theorem and illustrating various possible types of behavior with output from simulations.

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