Extremal distance and conformal radius of a CLE_4 loop

TL;DR

This paper computes the joint distribution of extremal distance and conformal radius for a CLE_4 loop in the unit disk, extending previous results and involving Brownian motion hitting times.

Contribution

It provides the first joint law of extremal distance and conformal radius for CLE_4 loops, using techniques involving Brownian motion.

Findings

Law of extremal distance between CLE_4 loop and boundary

Joint law of conformal radius and extremal distance involving Brownian motion

Joint laws of extremal distances in Brownian loop-soup clusters

Abstract

Consider CLE$_4$ in the unit disk and let $\ell$ be the loop of the CLE$_4$ surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by $\ell$. We complement their result by determining the law of the extremal distance between $\ell$ and the boundary of the unit disk. More surprisingly, we also compute the joint law of these conformal radius and extremal distance. This law involves first and last hitting times of a one-dimensional Brownian motion. Similar techniques also allow us to determine joint laws of some extremal distances in a critical Brownian loop-soup cluster.