Some Properties of the Plaquette Random-Cluster Model

TL;DR

This paper investigates the duality and boundary properties of the plaquette random-cluster model across different dimensions, providing new algebraic topology-based proofs for known results.

Contribution

It introduces algebraic topology techniques to analyze duality, boundary conditions, and limits in the plaquette random-cluster model, extending existing theoretical understanding.

Findings

Duality between i-dimensional and (d-i)-dimensional models established

Boundary conditions and infinite volume limits characterized

New algebraic topology proofs for existing results provided

Abstract

We show that the $i$-dimensional plaqutte random-cluster model with coefficients in $\mathbb{Z}_q$ is dual to a $(d-i)$-dimensional plaquette random cluster model. In addition, we explore boundary conditions, infinite volume limits, and uniqueness for these models. For previously known results, we provide new proofs that rely more on the tools of algebraic topology.