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Limiting shape for directed percolation models | | James B. Martin
; | | Date: |
7 Dec 2002 | | Journal: | Annals of Probability 2004, Vol. 32, No. 4, 2908-2937 DOI: 10.1214/009117904000000838 | | Subject: | Probability MSC-class: 60K35 (Primary) 82B43. (Secondary) | math.PR | | Abstract: | We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, dgeq2, with i.i.d. weights at the vertices. Under certain moment conditions on the common distribution of the weights, the limits g(x)=lim_{n oinfty}n^{-1}T(lfloor nx
floor) exist and are constant a.s. for xin R_+^d, where T(z) is the passage time from the origin to the vertex zin 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. | | Source: | arXiv, math.PR/0301055 | | Services: | Forum | Review | PDF | Favorites | |
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