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

@article{Dembin2018ExistenceOT,
title={Existence of the anchored isoperimetric profile in supercritical bond percolation in dimension two and higher},
author={Barbara Dembin},
journal={arXiv: Probability},
year={2018}
}
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… Expand
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