The Annals of Probability

Limiting shape for directed percolation models

James B. Martin

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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.

Article information

Ann. Probab., Volume 32, Number 4 (2004), 2908-2937.

First available in Project Euclid: 8 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Directed percolation first passage last passage shape theorem growth model


Martin, James B. Limiting shape for directed percolation models. Ann. Probab. 32 (2004), no. 4, 2908--2937. doi:10.1214/009117904000000838.

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