Graphical Representations for Ising and Potts Models in General External Fields

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].

56 pages. Accepted for publication in Journal of Statistical Physics

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• Stability of the Phase Transition of Critical-Field Ising Model on Cayley trees under Inhomogeneous External Fields
• Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields
• Absence of Replica Symmetry Breaking in Finite Fifth Moment Random Field Ising Model
• Thermodynamic Formalism for Generalized Markov Shifts on Infinitely Many States
• Sharp Phase Transition for the Random-Cluster Model with Summable External Magnetic Field