Limiting shape for directed percolation models

Details Limiting shape for directed percolation models Limiting shape for directed percolation models

2003-01-07

arxiv.org/abs/math/0301055v4

Annals of Probability 2004, Vol. 32, No. 4, 2908-2937

Mathematics

Probability

Published at http://dx.doi.org/10.1214/009117904000000838 in the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117904000000838

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z_+^d, d\geq2, 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\to\infty}n^{-1}T(\lfloor nx\rfloor) exist and are constant a.s. for x\in R_+^d, where T(z) is the passage time from the origin to the vertex z\in 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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