The time constant for Bernoulli percolation is Lipschitz continuous strictly above pc

Abstract

We consider the standard model of i.i.d. first passage percolation on ${\mathbb{Z}}^{\mathit{d}}$ given a distribution G on $[0,+\infty ]$ ($+\infty$ is allowed). When $\mathit{G}([0,+\infty ))>{\mathit{p}}_{\mathit{c}}(\mathit{d})$, it is known that the time constant ${\mathit{\mu }}_{\mathit{G}}$ exists. We are interested in the regularity properties of the map $\mathit{G}\mapsto {\mathit{\mu }}_{\mathit{G}}$. We first study the specific case of distributions of the form ${\mathit{G}}_{\mathit{p}}=\mathit{p}{\mathit{\delta }}_1+(1-\mathit{p}){\mathit{\delta }}_{\infty }$ for $\mathit{p}>{\mathit{p}}_{\mathit{c}}(\mathit{d})$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter p. We show that the function $\mathit{p}\mapsto {\mathit{\mu }}_{{\mathit{G}}_{\mathit{p}}}$ is Lipschitz continuous on every interval $[{\mathit{p}}_0,1]$, where ${\mathit{p}}_0>{\mathit{p}}_{\mathit{c}}(\mathit{d})$.