BOUNDARY TOUCHING PROBABILITY AND NESTED-PATH EXPONENT FOR NONSIMPLE CLE

• 成果类型: Article
• 署名作者: Ang, Morris; Sun, Xin; Yu, Pu; Zhuang, Zijie
• 署名单位: University of California System; University of California San Diego; Peking University; Massachusetts Institute of Technology (MIT); University of Pennsylvania
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1722
• 发表日期: 2025
• 页码: 797-847
• 关键词: liouville quantum-gravity; conformal-invariance; critical percolation; SURFACES; PERSPECTIVES; geometry; matings; MODEL; sle

摘要

The conformal loop ensemble (CLE) has two phases: for kappa E (8/3, 4], the loops are simple and do not touch each other or the boundary; for kappa E (4, 8), the loops are nonsimple and may touch each other and the boundary. For kappa E (4, 8), we derive the probability that the loop surrounding a given point touches the domain boundary. We also obtain the law of the conformal radius of this loop seen from the given point conditioned on the loop touching the boundary or not, refining a result of Schramm-Sheffield-Wilson (Comm. Math. Phys. (2009) 288 43-53). As an application, we exactly evaluate the CLE counterpart of the nested-path exponent for the Fortuin-Kasteleyn (FK) random cluster model recently introduced by Song-Tan-Zhang-Jacobsen-Nienhuis-Deng (J. Phys. A 55 (2022) Paper No. 204002). This exponent describes the asymptotic behavior of the number of nested open paths in the open cluster containing the origin when the cluster is large. For Bernoulli percolation, which corresponds to kappa = 6, the exponent was derived recently in Song-Jacobsen-Nienhuis-Sportiello-Deng (2023) by a color switching argument. For kappa not equal 6 and, in particular, for the FK-Ising case, our formula appears to be new. Our derivation begins with Sheffield's construction of CLE from which the quantities of interest can be expressed by radial SLE. We solve the radial SLE problem using the coupling between SLE and Liouville quantum gravity, along with the exact solvability of Liouville conformal field theory.