Year: 2025
Type: preprint
Abstract: Recently, Ang--Cai--Sun--Wu (2024) determined the three-point connectivity constant for two-dimensional critical percolation, confirming a prediction of Delfino and Viti (2010). In this paper, we address the analogous problem for planar critical $q$-state Potts spin clusters. We introduce a continuum three-point connectivity constant and compute it explicitly. Under the scaling-limit conjecture for Potts spin clusters, this quantity coincides with the scaling limit of the properly normalized probability that three points lie in the same spin cluster. The resulting formula agrees with the imaginary DOZZ formula up to an explicit $q$-dependent constant with a geometric interpretation. This answers a question from Delfino--Picco--Santachiara--Viti (2013). The proof exploits the coupling between CLE and LQG, together with the BCLE descriptions of $q$-state Potts scaling limits due to Miller--Sheffield--Werner (2017) and Köhler-Schindler and Lehmkühler (2025).
Source: arXiv (Cornell University)
Authors: Cai, Gefei, Liu, Haoyu, Wu, Baojun, Zhuang, Zijie
Institutions: Peking University, University of Pennsylvania
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4414979587
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