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

TL;DR

This paper establishes a lower bound on the difference of the time constants in Bernoulli first-passage percolation on integer lattices, using Russo's formula to analyze how the time constant varies with the parameter p.

Contribution

It provides a novel lower bound for the difference in time constants for different p-values, advancing understanding of the model's behavior.

Findings

Proves a lower bound for the difference of time constants using Russo's formula.

Shows the monotonicity and bounds of the time constant in Bernoulli first-passage percolation.

Provides a mathematical inequality relating the time constants at different p-values.

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.