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

Abstract

We consider the standard model of first-passage percolation on ${\mathbb{Z}}^{\mathit{d}}$ ($\mathit{d}\ge 2$), 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 $1-\mathit{\epsilon }$. These passage times induce a random pseudo-metric ${\mathit{T}}_{\mathit{\epsilon }}$ on ${\mathbb{R}}^{\mathit{d}}$. By subadditive arguments, it is well known that for any $\mathit{z}\in {\mathbb{R}}^{\mathit{d}}\setminus \{0\}$, the sequence ${\mathit{T}}_{\mathit{\epsilon }}(0,\mathit{n}\mathit{z})/\mathit{n}$ converges a.s. toward a constant ${\mathit{\mu }}_{\mathit{\epsilon }}(\mathit{z})$ called the time constant. We investigate the behavior of $\mathit{\epsilon }\mapsto {\mathit{\mu }}_{\mathit{\epsilon }}(\mathit{z})$ near 0, and prove that ${\mathit{\mu }}_{\mathit{\epsilon }}(\mathit{z})=\Vert \mathit{z}{\Vert }_1-\mathit{C}(\mathit{z}){\mathit{\epsilon }}^{1/{\mathit{d}}_1(\mathit{z})}+\mathit{o}({\mathit{\epsilon }}^{1/{\mathit{d}}_1(\mathit{z})})$, where ${\mathit{d}}_1(\mathit{z})$ is the number of nonnull coordinates of z, and $\mathit{C}(\mathit{z})$ is a constant whose dependence on z is partially explicit.