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

## TL;DR

This paper proves that the anchored isoperimetric profile of the infinite cluster in supercritical bond percolation on  lattices varies in a Lipschitz continuous manner with respect to the percolation parameter p, for all p above the critical threshold.

## Contribution

It establishes the Lipschitz continuity of the anchored isoperimetric profile in the supercritical regime, a novel regularity result for percolation clusters.

## Key findings

- Lipschitz continuity holds for the isoperimetric profile in the supercritical phase
- Continuity is valid across all p in (p_c(d), 1)
- Results apply to all dimensions d  2

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