## 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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### References:

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