FIRST-ORDER BEHAVIOR OF THE TIME CONSTANT IN BERNOULLI FIRST-PASSAGE PERCOLATION

• 成果类型: Article
• 署名作者: Basdevant, Anne-Laure; Gouere, Jean-Baptiste; Theret, Marie
• 署名单位: Universite de Tours
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/22-AAP1795
• 发表日期: 2022
• 页码: 4535-4567
• 关键词: passage; lines

摘要

We consider the standard model of first-passage percolation on Z(d) (d >= 2), with i.i.d. passage times associated with either the edges or the vertices of the graph. We focus on the particular case where the distribution of the passage times is the Bernoulli distribution with parameter 1 - epsilon. These passage times induce a random pseudo-metric T-epsilon on R-d. By subadditive arguments, it is well known that for any z is an element of R-d \ {0}, the sequence T-epsilon(0, nz)/n converges a.s. toward a constant mu(epsilon)(z) called the time constant. We investigate the behavior of epsilon (sic) mu(epsilon)(z) near 0, and prove that mu(epsilon)(z) = parallel to z parallel to(1) - C(z)epsilon(1/d1(z)) + o(epsilon(1/d1(z))), where d(1)(z) is the number of nonnull coordinates of z, and C(z) is a constant whose dependence on z is partially explicit.