2003/01/31 by Jean Martin, James B. Martin · 7 citations
Mathematics · Physics and Astronomy · #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #msc:60K35 #msc:82B43.
paper · pdf · doi:10.1214/009117904000000838
published as Annals of Probability 2004, Vol. 32, No. 4, 2908-2937 · Published at http://dx.doi.org/10.1214/009117904000000838 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2004/10/01 · arxiv created 2005/04/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.