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First-order behavior of the time constant in Bernoulli first-passage percolation

Abstract : 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03267940
Contributor : Marie Théret <>
Submitted on : Tuesday, June 22, 2021 - 5:31:52 PM
Last modification on : Tuesday, July 13, 2021 - 3:43:33 AM

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  • HAL Id : hal-03267940, version 1
  • ARXIV : 2106.12266

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Anne-Laure Basdevant, Jean-Baptiste Gouéré, Marie Théret. First-order behavior of the time constant in Bernoulli first-passage percolation. 2021. ⟨hal-03267940⟩

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