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

Barbara Dembin

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arXiv · 1901.00367

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

Barbara Dembin

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Abstract

We consider an i.i.d. supercritical bond percolation on $\mathbb{Z}^d$, every edge is open with a probability $p > p_c (d)$, where $p_c (d)$ denotes the critical parameter for this percolation. We know that there exists almost surely a unique infinite open cluster $C_p$ [7]. We are interested in the regularity properties in p of the anchored isoperimetric profile of the infinite cluster $C_p$. For $d\ge2$, we prove that the anchored isoperimetric profile defined in [4] is Lipschitz continuous on all intervals $[p_0 , p_1 ] \subset (p_c (d), 1)$.

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Barbara Dembin. 2018-12-29. Anchored isoperimetric profile of the infinite cluster in supercritical bond percolation is Lipschitz continuous. https://arxiv.org/abs/1901.00367

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