Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance

TL;DR

This paper conjectures exact formulas for four-point connectivities in critical 2D Potts models, including percolation, and confirms them with Monte Carlo simulations, revealing universal behavior and singularities.

Contribution

It introduces conjectured exact expressions for four-point connectivities in 2D Potts models and percolation, supported by numerical validation.

Findings

Excellent agreement between conjectures and simulations for Q=1,2,3

Universal ratios of connectivities are identified

Logarithmic singularity observed in percolation limit

Abstract

We conjecture an exact form for an universal ratio of four-point cluster connectivities in the critical two-dimensional $Q$-color Potts model. We also provide analogous results for the limit $Q\rightarrow 1$ that corresponds to percolation where the observable has a logarithmic singularity. Our conjectures are tested against Monte Carlo simulations showing excellent agreement for $Q=1,2,3$.