On a Lower Bound for the Time Constant of First-Passage Percolation

Abstract

We consider the Bernoulli first-passage percolation on $\mathbb Z^d (d\ge 2)$. That is, the edge passage time is taken independently to be 1 with probability $1-p$ and 0 otherwise. Let ${\mu(p)}$ be the time constant. We prove in this paper that \[ \mu(p_1)-\mu({p_2})\ge \frac{\mu(p_2)}{1-p_2}(p_2-p_1)\] for all $ 0\leq p_1<p_2< 1$ by using Russo's formula.