On the crossing estimates for simple conformal loop ensembles

TL;DR

This paper establishes the super-exponential decay of crossing probabilities in simple conformal loop ensembles, aiding the proof of convergence of double-dimer loop ensembles to nested CLE4.

Contribution

It proves a key decay estimate for crossing probabilities in CLE, filling a gap in the understanding of double-dimer loop ensemble convergence.

Findings

Super-exponential decay of crossing probabilities in CLE

Provides a crucial ingredient for convergence proofs of double-dimer ensembles

Enhances understanding of crossing behavior in conformal loop ensembles

Abstract

We prove the super-exponential decay of probabilities that there exist $n$ crossings of a given quadrilateral in a simple $\text{CLE}_\kappa(\Omega)$, $\frac{8}{3}<\kappa\le 4$, as $n$ goes to infinity. Besides being of independent interest, this also provides the missing ingredient in arXiv:1809.00690 for proving the convergence of probabilities of cylindrical events for the double-dimer loop ensembles to those for the nested $\text{CLE}_4(\Omega)$.