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

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