## View abstract

### Session S04 - Random Walks and Related Topics

Thursday, July 15, 14:20 ~ 14:50 UTC-3

## Extremal distance and conformal radius of a $CLE(4)$ loop

### Avelio Sepúlveda

#### Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak75d4fe08e5323ce39ed3fb9ed9aef175').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy75d4fe08e5323ce39ed3fb9ed9aef175 = 'ls&#101;p&#117;lv&#101;d&#97;' + '&#64;'; addy75d4fe08e5323ce39ed3fb9ed9aef175 = addy75d4fe08e5323ce39ed3fb9ed9aef175 + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl'; var addy_text75d4fe08e5323ce39ed3fb9ed9aef175 = 'ls&#101;p&#117;lv&#101;d&#97;' + '&#64;' + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl';document.getElementById('cloak75d4fe08e5323ce39ed3fb9ed9aef175').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy75d4fe08e5323ce39ed3fb9ed9aef175 + '\'>'+addy_text75d4fe08e5323ce39ed3fb9ed9aef175+'<\/a>';

In this talk, we will discuss the geometry of the loop of a $CLE(4)$ surrounding the origin. In particular, we show how to compute the joint law between the conformal radius seen from the origin of the domain surrounded by the loop together with the extremal distance from this loop to the boundary. This joint law is related to certain random times of the Brownian motion, more precisely the first exit time of the set $[-2\pi, 2\pi]$ together with the last hit of $0$. We will explain where this relationship comes from together with a new characterization of this coupling.

Joint work with Juhan Aru (École polytechnique fédérale de Lausanne, Switzerland) and Titus Lupu (CNRS, France).