The square summability of the CLE complementary component diameters

Abstract

We show that the sum of the squares of the diameters of the complementary connected components of the CLE$_\kappa$ carpet/gasket is almost surely finite for $\kappa \in (8/3, 4) \cup (4, 8)$. This is a prerequisite for the application of a result of Ntalampekos which allows the CLE$_\kappa$ carpet/gasket to be uniformized to a round Sierpi\'nski packing, in analogy with the classical Koebe uniformization theorem for finitely connected domains. Our result is new in the case that $\kappa \in (4,8)$ and we provide a new proof for $\kappa \in (8/3, 4)$. In both cases we use the link between CLE and space-filling SLE. The square-summability of diameters has been proved for $\kappa \in (8/3, 4]$ in unpublished work by Rohde and Werness using a different method. Our work completes the proof that this property holds for all $\kappa$ for which CLE$_\kappa$ is defined.