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Graphical Representations for Ising and Potts Models in General External Fields

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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-state 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 almost sure 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 with small enough power, as conjectured in Bissacot et al. (Commun Math Phys 337: 41–53, 2015).

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Notes

  1. in case \(\beta _c(\varvec{J},\varvec{h}) = +\infty \) we mean that the set of the Gibbs measures is a singleton for all \(0<\beta <+\infty \).

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Acknowledgments

The authors thank Aernout van Enter for fruitful discussions. We also acknowledge the anonymous referees for their helpful comments, suggestions and references provided in their reports. Leandro Cioletti is supported by FEMAT and Roberto Vila is supported by CNPq.

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  1. Departamento de Matemática, Universidade de Brasília, Brasília, Brazil

    Leandro Cioletti & Roberto Vila

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  1. Leandro Cioletti
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  2. Roberto Vila
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Correspondence to Leandro Cioletti.

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Cioletti, L., Vila, R. Graphical Representations for Ising and Potts Models in General External Fields. J Stat Phys 162, 81–122 (2016). https://doi.org/10.1007/s10955-015-1396-5

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