Limiting shape for directed percolation models

• 成果类型: Article
• 署名作者: Martin, JB
• 署名单位: Universite Paris Cite; Centre National de la Recherche Scientifique (CNRS); Universite Paris Cite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000838
• 发表日期: 2004
• 页码: 2908-2937
• 关键词: 1st passage percolation; tandem queuing-networks; greedy lattice animals; 1st-passage percolation; RANDOM MATRICES; fluctuations; GROWTH; dimensions; BEHAVIOR; THEOREMS

摘要

We consider directed first-passage and last-passage percolation on the nonnegative lattice Z(+)(d), d greater than or equal to 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-->infinity)n(-1)T([nx]) exist and are constant a.s. for x is an element of R-+(d), where T(z) is the passage time from the origin to the vertex z is an element of 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.