Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous

Abstract

We consider the standard model of i.i.d. first passage percolation on $\mathbb {Z}^{d}$ given a distribution $G$ on $\mathbb {R}_{+}$. We consider a cube oriented in the direction $\overrightarrow {v}$ whose sides have length $n$. We study the maximal flow from the top half to the bottom half of the boundary of this cube. We already know that the maximal flow renormalized by $n^{d-1}$ converges towards the flow constant $\nu _{G}(\overrightarrow {v})$. We prove here that the map $p\mapsto \nu _{p\delta _{1}+(1-p)\delta _{0}}$ is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$ where $p_{c}(d)$ denotes the critical parameter for i.i.d. bond percolation on $\mathbb {Z}^{d}$. For $p>p_{c}(d)$, we know that there exists almost surely a unique infinite open cluster $\mathcal {C}_{p}$ [8]. We are interested in the regularity properties in $p$ of the anchored isoperimetric profile of the infinite cluster $\mathcal {C}_{p}$. For $d\geq 2$, using the result on the regularity of the flow constant, we prove here that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals $[p_{0},p_{1}]\subset (p_{c}(d),1)$.