Extreme nesting in the conformal loop ensemble

Abstract

The conformal loop ensemble $\operatorname {CLE}_κ$ with parameter $8/3<κ<8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $κ$ and $ν$, we compute the almost-sure Hausdorff dimension of the set of points $z$ for which the number of CLE loops surrounding the disk of radius $\varepsilon$ centered at $z$ has asymptotic growth $ν\log (1/\varepsilon )$ as $\varepsilon \to0$. By extending these results to a setting in which the loops are given i.i.d. weights, we give a CLE-based treatment of the extremes of the Gaussian free field.