Abstract
Consider in the unit disk, and let ℓ be the loop of the surrounding the origin. Schramm, Sheffield and Wilson determined the law of the conformal radius seen from the origin of the domain surrounded by ℓ. We complement their result by determining the law of the extremal distance between ℓ 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.
Funding Statement
This work was partially supported by the SNF Grants SNF-155922 and SNF-175505. J. Aru is a member of NCCR Swissmap and is now supported by SNF Eccellenza grant 194648. T. Lupu acknowledges the support of the French National Research Agency (ANR) grant within the project MALIN (ANR-16-CE93-0003). A. Sepúlveda was supported by the ERC grant LiKo 676999 and is now supported by grant ANID ACE210010, fondo basal AFB210005 and FONDECYT iniciación de investigación 11200085.
Acknowledgments
The authors are thankful to Wendelin Werner for inspiring discussions and for pointing out that the joint laws of multiple nested interfaces cannot be read out from a Brownian motion in the naive way. Furthermore, we would like to thank an anonymous referee for important comments on an earlier version of this paper. A. Sepúlveda would also like to thank the hospitality of Núcleo Milenio “Stochastic models of complex and disordered systems” for repeated invitations to Santiago, where part of this paper was written.
Citation
Juhan Aru. Titus Lupu. Avelio Sepúlveda. "Extremal distance and conformal radius of a loop." Ann. Probab. 50 (2) 509 - 558, March 2022. https://doi.org/10.1214/21-AOP1538
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