Continuity of the time and isoperimetric constants in supercritical percolation

TL;DR

This paper proves the continuity of the isoperimetric constant and the time constant in supercritical percolation models, showing stability of these geometric and probabilistic quantities under changes in the environment law.

Contribution

It establishes the continuity of the isoperimetric and time constants in supercritical percolation, extending previous results to include possibly infinite passage times and the isoperimetric constant.

Findings

Isoperimetric constant is continuous in the percolation parameter in $ ext{Z}^2$.

Time constant is continuous with respect to the law of passage times.

Normalized sets achieving the isoperimetric constant vary continuously in the Hausdorff metric.

Abstract

We consider two different objects on super-critical Bernoulli percolation on $\mathbb{Z}^d$ : the time constant for i.i.d. first-passage percolation (for $d\geq 2$) and the isoperimetric constant (for $d=2$). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in $\mathbb{Z}^2$ is continuous in the percolation parameter. As a corollary we prove that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdroff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on $\mathbb{Z}^d$ with possibly infinite passage times: we associate with each edge $e$ of the graph a passage time $t(e)$ taking values in $[0,+\infty]$, such that $\mathbf{P}[t(e)<+\infty] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property previously proved by Cox and Kesten for first passage percolation with finite passage times.