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

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 $0\le p_1<p_2<1$ by using Russo’s formula.