Extreme nesting in the conformal loop ensemble

The conformal loop ensemble ${\mathrm{CLE}}_{\kappa }$ with parameter $8\mathrm{/}3<\kappa <8$ is the canonical conformally invariant measure on countably infinite collections of noncrossing loops in a simply connected domain. Given $\kappa$ and $\nu$, 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 $\epsilon$ centered at $z$ has asymptotic growth $\nu \log (1\mathrm{/}\epsilon )$ as $\epsilon \to 0$. 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.