Boundary four-point connectivities of conformal loop ensembles

TL;DR

This paper derives explicit boundary four-point Green's functions for conformal loop ensembles with specific parameters, confirming conjectures in percolation and FK-Ising models and extending existing factorization formulas.

Contribution

It provides exact formulas for boundary connectivities in CLE for ppa in (4,8), including special cases and singularities, extending previous factorization results.

Findings

Established exact boundary four-point connectivities for ppa=6 and 16/3.

Confirmed conjectures for critical Bernoulli percolation and FK-Ising models.

Identified a logarithmic singularity in the FK-Ising model.

Abstract

We derive the boundary four-point Green's functions for conformal loop ensembles (CLE) with $\kappa\in(4,8)$. Specializing to $\kappa=6$ and $\kappa=16/3$, we establish the exact formulas for the boundary four-point connectivities in critical Bernoulli percolation and the FK-Ising model conjectured by Gori-Viti (2017, 2018). In particular, we identify a logarithmic singularity in the critical FK-Ising model. Our approach also applies to the one-bulk and two-boundary connectivities of CLE, thereby extending the factorization formula of Beliaev-Izyurov (2012) to all $\kappa\in(4,8)$.