Graphical Representations for Ising and Potts Models in General External Fields

TL;DR

This paper develops a comprehensive graphical representation framework for Ising and Potts models on general lattices with non-uniform external fields, deriving key properties and proving measure uniqueness under certain conditions.

Contribution

It introduces a detailed Random Cluster Representation for these models with non-translation invariant fields and establishes measure properties like FKG inequality and uniqueness of Gibbs measures.

Findings

Explicit distribution functions for models with general external fields

Proved FKG inequality for the GRC Model in non-translation invariant case

Established uniqueness of Gibbs measures for ferromagnetic Ising model with power-law decay magnetic field

Abstract

This work is concerned with the theory of Graphical Representation for the Ising and Potts Models over general lattices with non-translation invariant external field. We explicitly describe in terms of the Random Cluster Representation the distribution function and, consequently, the expected value of a single spin for the Ising and $q$-states Potts Models with general external fields. We also consider the Gibbs States for the Edwards-Sokal Representation of the Potts Model with non-translation invariant magnetic field and prove a version of the FKG Inequality for the so called General Random Cluster Model (GRC Model) with free and wired boundary conditions in the non-translation invariant case. Adding the amenability hypothesis on the lattice, we obtain the uniqueness of the infinite connected component and the quasilocality of the Gibbs Measures for the GRC Model with such general magnetic fields. As a final application of the theory developed, we show the uniqueness of the Gibbs Measures for the Ferromagnetic Ising Model with a positive power law decay magnetic field, as conjectured in [8].