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Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous. (English) Zbl 1434.60282

Summary: 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 \([p_0,p_1]\subset (p_c(d),1)\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
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