Random Cluster Models on the Triangular Lattice

TL;DR

This paper investigates percolation and the random cluster model on the triangular lattice with innovative 3-body interactions, establishing conditions for phase transitions and correlations using generalized transformations.

Contribution

It introduces a new parameter for 3-body interactions, generalizes the star-triangle transformation, and derives duality and correlation conditions for the random cluster model.

Findings

Identified conditions for positive correlations in the model

Established regions of percolation and non-percolation

Proved theorems on phase transitions

Abstract

We study percolation and the random cluster model on the triangular lattice with 3-body interactions. Starting with percolation, we generalize the star--triangle transformation: We introduce a new parameter (the 3-body term) and identify configurations on the triangles solely by their connectivity. In this new setup, necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non-percolation. Next we apply this set of ideas to the $q>1$ random cluster model: We derive duality relations for the suitable random cluster measures, prove necessary and sufficient conditions for them to have positive correlations, and finally prove some rigorous theorems concerning phase transitions.