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

TL;DR

This paper analyzes how the time constant in Bernoulli first-passage percolation on b^d behaves as the Bernoulli parameter approaches 1, revealing a first-order asymptotic expansion in terms of psilon.

Contribution

It provides a novel asymptotic expansion of the time constant near psilon=0, detailing the dependence on the number of non-zero coordinates of the vector z.

Findings

The time constant psilon(z) approaches the -norm of z as psilon.

A precise asymptotic expansion of psilon(z) in terms of psilon is established.

The exponent in the expansion depends on the number of non-zero coordinates of z.

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.