Continuity of the time and isoperimetric constants in supercritical percolation

Abstract

We consider two different objects on supercritical Bernoulli percolation on the edges of $\mathbb{Z} ^d$ : 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 obtain that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdorff 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 $\mathbb{P} [t(e)<+\infty ] >p_c(d)$. We prove the continuity of the time constant with respect to the law of the passage times. This extends the continuity property of the asymptotic shape previously proved by Cox and Kesten [8, 10, 20] for first-passage percolation with finite passage times.