Extreme nesting in the conformal loop ensemble

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