*Published Paper*

**Inserted:** 10 oct 2022

**Last Updated:** 23 apr 2023

**Journal:** Networks and Heterogeneous Media

**Volume:** 18

**Number:** 3

**Pages:** 1226-1235

**Year:** 2023

**Doi:** 10.3934/nhm.2023053

**Abstract:**

An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is the existence of a cluster $\mathbf E$ which has finite and minimal perimeter $P(\mathbf E)$ among all clusters with regions having such volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbf E)$, where $\partial \mathbf E$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.

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