# 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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