# Extremal distance and conformal radius of a CLE_4 loop

Juhan Aru, Titus Lupu, Avelio Sepúlveda

Juhan Aru, Titus Lupu, Avelio Sepúlveda

**28/4/20**Published in : arXiv:2004.13570

Consider CLE_4 in the unit disk and let /eli 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 /eli. We complement their result by determining the law of the extremal distance between /eli 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.