On the Crossing Estimates for Simple Conformal Loop Ensembles

Abstract

Abstract We prove a super-exponential decay of probabilities that there exist $n$ crossings of a given quad for a simple $\textrm {CLE}_\kappa (\Omega )$, $\frac {8}{3}<\kappa \le 4$, as $n$ goes to infinity. Besides, being of independent interest, this also provides the missing ingredient in [2] for proving the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested $\textrm {CLE}_4(\Omega )$.