On Convergence to SLE$_6$ I: Conformal Invariance for Certain Models of the Bond-Triangular Type

TL;DR

This paper proves that certain bond-triangular models of percolation in two dimensions converge to SLE$_6$, establishing their conformal invariance and universality class, under broad conditions including general domains and criticality.

Contribution

It extends convergence results to SLE$_6$ for correlated bond-triangular models, broadening the class of models known to exhibit conformal invariance at criticality.

Findings

Models are in the same universality class as standard site percolation on the triangular lattice.

Convergence to SLE$_6$ is proven for all domains with boundary Minkowski dimension less than two.

The proof applies to general critical 2D percolation models assuming Cardy's Formula holds.

Abstract

Following the approach outlined in [26], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [11] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [27]. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy's Formula can be established for domains in this generality.