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 \[ μ(p_1)-μ({p_2})\ge \frac{μ(p_2)}{1-p_2}(p_2-p_1)\] for all $0\le p_1<p_2<1$ by using Russo's formula.

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