Three-point connectivity constant for $q$-state Potts spin clusters

TL;DR

This work defines and computes the continuum three-point connectivity constant for planar critical $q$-state Potts spin clusters, linking it to the imaginary DOZZ formula via a $q$-dependent normalization. The authors leverage the Edwards–Sokal coupling, conformal loop ensembles (CLE/BCLE), and Liouville quantum gravity (LQG) to relate fuzzy Potts interfaces to CLE loop measures, establishing a universal constant $\mathsf{C}(\kappa)$ that bridges fuzzy Potts and CLE. The main result expresses the connectivity constant as $R(q)=C(q)\cdot C^{\rm ImDOZZ}_{\beta}(\frac{1}{4\beta}-\frac{\beta}{2},\frac{1}{4\beta}-\frac{\beta}{2},\frac{1}{4\beta}-\frac{\beta}{2})$, with $\beta=2/\sqrt{\kappa}$ and $\kappa=4\arccos(-\sqrt{q}/2)/\pi$, and provides an explicit evaluation of $\mathsf{C}(\kappa)$ as $\mathsf{C}(\kappa)=\sqrt{\kappa/2}\,\sin(\kappa\pi/2)/\sin(4\pi/\kappa)$. The approach yields a rigorous probabilistic framework for multipoint connectivity, confirming predictions from conformal field theory and aligning with numerical results for $q=3$ and percolation limits. This bridges probabilistic loop models with CFT structure constants, enriching the understanding of Potts model scaling limits and their geometric-analytic content.

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).

Theorems & Definitions (29)

• Definition 1.1
• Theorem 1.2
• Conjecture 1.3
• Remark 1.4
• Theorem 1.5
• Proposition 1.6
• Theorem 2.1
• Theorem 3.1
• Theorem 3.2
• proof : Proof of Theorem \ref{['thm: main']}