Extreme nesting in the conformal loop ensemble

Abstract

The conformal loop ensemble ${\mathrm{CLE}}_{\kappa }$ with parameter $8/3<\kappa <8$ 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 $\nu \log (1/\epsilon )$ 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.

Published at http://dx.doi.org/10.1214/14-AOP995 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• A Guide to Stochastic Loewner Evolution and its Applications
• Two-Dimensional Critical Percolation: The Full Scaling Limit
• Conformal radii for conformal loop ensembles

Cited by in corpus (2)

• The dimension of the boundary of a Liouville quantum gravity metric ball
• Nesting statistics in the O(n) loop model on random planar maps