Boundary touching probability and nested-path exponent for non-simple CLE

TL;DR

This paper analyzes the probability of boundary touching by loops in non-simple conformal loop ensembles (CLE) for rom 4 to 8, deriving new formulas for nested-path exponents relevant to FK models and percolation.

Contribution

It provides the first explicit formulas for boundary touching probabilities and nested-path exponents in non-simple CLE, extending previous results to rom 4 to 8.

Findings

Derived boundary touching probability for non-simple CLE.

Obtained the law of conformal radius conditioned on boundary touching.

Explicitly evaluated the nested-path exponent for FK models.

Abstract

The conformal loop ensemble (CLE) has two phases: for $\kappa \in (8/3, 4]$, the loops are simple and do not touch each other or the boundary; for $\kappa \in (4,8)$, the loops are non-simple and may touch each other and the boundary. For $\kappa\in(4,8)$, we derive the probability that the loop surrounding a given point touches the domain boundary. We also obtain the law of the conformal radius of this loop seen from the given point conditioned on the loop touching the boundary or not, refining a result of Schramm-Sheffield-Wilson (2009). As an application, we exactly evaluate the CLE counterpart of the nested-path exponent for the Fortuin-Kasteleyn (FK) random cluster model recently introduced by Song-Tan-Zhang-Jacobsen-Nienhuis-Deng (2022). This exponent describes the asymptotic behavior of the number of nested open paths in the open cluster containing the origin when the cluster is large. For Bernoulli percolation, which corresponds to $\kappa=6$, the exponent was derived recently in Song-Jacobsen-Nienhuis-Sportiello-Deng (2023) by a color switching argument. For $\kappa\neq 6$, and in particular for the FK-Ising case, our formula appears to be new. Our derivation begins with Sheffield's construction of CLE from which the quantities of interest can be expressed by radial SLE. We solve the radial SLE problem using the coupling between SLE and Liouville quantum gravity, along with the exact solvability of Liouville conformal field theory.