1. First-order behavior of the time constant in Bernoulli first-passage percolation
- Author
-
Basdevant, Anne-Laure, Gouéré, Jean-Baptiste, Théret, Marie, Fédération Parisienne de Modélisation Mathématique (FP2M), Centre National de la Recherche Scientifique (CNRS), Modélisation aléatoire de Paris X (MODAL'X), Université Paris Nanterre (UPN), Institut Denis Poisson (IDP), Centre National de la Recherche Scientifique (CNRS)-Université de Tours-Université d'Orléans (UO), Labex MME-DII (ANR 11-LBX-0023-01)., ANR-16-CE40-0016,PPPP,Percolation et percolation de premier passage(2016), Université d'Orléans (UO)-Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-Université de Tours (UT)-Université d'Orléans (UO)
- Subjects
Statistics and Probability ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Probability (math.PR) ,FOS: Mathematics ,Statistics, Probability and Uncertainty ,Mathematics - Probability - Abstract
28 pages, 16 ref.; We consider the standard model of first-passage percolation on $\mathbb{Z}^d$ ($d\geq 2$), 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 $\mathbb{R}^d$. By subadditive arguments, it is well known that for any $z\in\mathbb{R}^d\setminus \{0\}$, the sequence $T_\epsilon (0,\lfloor nz \rfloor) / n$ converges a.s. towards a constant $\mu_\epsilon (z)$ called the time constant. We investigate the behavior of $\epsilon \mapsto \mu_\epsilon (z)$ near $0$, and prove that $\mu_\epsilon (z) = \| z\|_1 - C (z) \epsilon ^{1/d_1(z)} + o ( \epsilon ^{1/d_1(z)}) $, where $d_1(z)$ is the number of non null coordinates of $z$, and $C(z)$ is a constant whose dependence on $z$ is partially explicit.
- Published
- 2021