Extremal distance and conformal radius of a CLE_4 loop

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.

45 pages, 8 figures

References in corpus (5)

• A Guide to Stochastic Loewner Evolution and its Applications
• Conformal radii for conformal loop ensembles
• The first passage sets of the 2D Gaussian free field: convergence and isomorphisms
• Lecture notes on the Gaussian Free Field
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Cited by in corpus (2)

• Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond
• An equivalence between gauge-twisted and topologically conditioned scalar Gaussian free fields