The time constant for Bernoulli percolation is Lipschitz continuous strictly above $p_c$

TL;DR

This paper proves that the time constant in Bernoulli first passage percolation is Lipschitz continuous for parameters strictly above the critical threshold, ensuring regularity in the model's behavior.

Contribution

It establishes the Lipschitz continuity of the time constant map for Bernoulli percolation models above the critical probability.

Findings

Lipschitz continuity of the time constant for p > p_c(d)

Regularity of the time constant map in Bernoulli percolation

Continuity results for the percolation model's parameters

Abstract

We consider the standard model of i.i.d. first passage percolation on $\mathbb{Z}^d$ given a distribution $G$ on $[0,+\infty]$ ($+\infty$ is allowed). When $G([0,+\infty]) < p_c(d)$, it is known that the time constant $\mu_G$ exists. We are interested in the regularity properties of the map $G\mapsto\mu_G$. We first study the specific case of distributions of the form $G_p=p\delta_1+(1-p)\delta_\infty$ for $p>p_c(d)$. In this case, the travel time between two points is equal to the length of the shortest path between the two points in a bond percolation of parameter $p$. We show that the function $p\mapsto \mu_{G_p}$ is Lipschitz continuous on every interval $[p_0,1]$, where $p_0>p_c(d)$.