Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher

TL;DR

This paper proves that in supercritical bond percolation on high-dimensional lattices, the minimal boundary-to-volume ratio subgraphs converge to a deterministic shape with a constant ratio, extending known results to higher dimensions.

Contribution

It establishes the existence of an anchored isoperimetric profile in supercritical bond percolation for all dimensions d ≥ 2, showing convergence to a deterministic shape and ratio.

Findings

Minimizers converge to a deterministic shape after rescaling

Boundary-to-volume ratio converges to a deterministic constant

Results hold in all dimensions d ≥ 2

Abstract

Let $d\geq 2$. 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 point. We condition on the event that $0$ belongs to the infinite cluster $\mathcal{C}_\infty$ and we consider connected subgraphs of $\mathcal{C}_\infty$ having at most $n^d$ vertices and containing $0$. Among these subgraphs, we are interested in the ones that minimize the open edge boundary size to volume ratio. These minimizers properly rescaled converge towards a translate of a deterministic shape and their open edge boundary size to volume ratio properly rescaled converges towards a deterministic constant.