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October 2004 Limiting shape for directed percolation models
James B. Martin
Ann. Probab. 32(4): 2908-2937 (October 2004). DOI: 10.1214/009117904000000838


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(x)=lim n→∞n−1T(⌊nx⌋) 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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James B. Martin. "Limiting shape for directed percolation models." Ann. Probab. 32 (4) 2908 - 2937, October 2004.


Published: October 2004
First available in Project Euclid: 8 February 2005

zbMATH: 1065.60149
MathSciNet: MR2094434
Digital Object Identifier: 10.1214/009117904000000838

Primary: 60K35
Secondary: 82B43

Keywords: directed percolation , first passage , Growth model , last passage , shape theorem

Rights: Copyright © 2004 Institute of Mathematical Statistics


Vol.32 • No. 4 • October 2004
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