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

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

2008-07-05

arxiv.org/abs/0807.0839v2

Mathematics

Probability

7 pages

Scientific paper

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

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