Potts q-color field theory and scaling random cluster model

TL;DR

This paper explores the structural properties of the Potts q-color field theory, linking it to the random cluster model, and investigates correlators, operator expansions, and duality relations in two dimensions.

Contribution

It establishes the equivalence of independent n-point correlators and cluster connectivities, and analyzes duality and scattering in 2D, providing new insights into the Potts model's structure.

Findings

Number of independent n-point correlators equals generalized Bell numbers.

Identified the operator product expansion structure for generic q.

Derived a sum rule for kink-kink elastic scattering amplitudes.

Abstract

We study structural properties of the q-color Potts field theory which, for real values of q, describes the scaling limit of the random cluster model. We show that the number of independent n-point Potts spin correlators coincides with that of independent n-point cluster connectivities and is given by generalized Bell numbers. Only a subset of these spin correlators enters the determination of the Potts magnetic properties for q integer. The structure of the operator product expansion of the spin fields for generic q is also identified. For the two-dimensional case, we analyze the duality relation between spin and kink field correlators, both for the bulk and boundary cases, obtaining in particular a sum rule for the kink-kink elastic scattering amplitudes.