The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$

Abstract

We consider the standard model of i.i.d. first passage percolation on ${\mathbb{Z}}^d$ given a distribution $G$ on $[0,+\infty ]$ ($+\infty$ is allowed). When $G([0,+\infty ])<p_c(d)$, it is known that the time constant $\hat{I}{¼}_G$ exists. We are interested in the regularity properties of the map $G\mapsto \hat{I}{¼}_G$. We first study the specific case of distributions of the form $G_p=p\hat{I}{´}_1+(1-p)\hat{I}{´}_{\infty }$ for $p>p_c(d)$. 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 $p\mapsto \hat{I}{¼}_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$.