New sum rule identities and duality relation for the Potts $n$-point correlation function

TL;DR

This paper derives new sum rule identities and duality relations for the Potts model's n-point correlation functions on planar lattices, advancing the theoretical understanding of these correlations.

Contribution

It introduces explicit sum rule identities for n=4,5 and establishes the duality relation for the n-point correlation function, filling gaps in the theoretical framework.

Findings

Explicit identities for n=4,5 correlation functions

Complete duality relation for the 4-point case

General conjecture on the form of duality relations

Abstract

It is shown that certain sum rule identities exist which relate correlation functions for $n$ Potts spins on the boundary of a planar lattice for $n\geq 4$. Explicit expressions of the identities are obtained for $n=4,5$. It is also shown that the identities provide the missing link needed for a complete determination of the duality relation for the $n$-point correlation function. The $n=4$ duality relation is obtained explicitly. More generally we deduce the number of correlation identities for any $n$ as well as an inversion relation and a conjecture on the general form of the duality relation.