  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.
