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March 2020 Translation-invariant Gibbs states of the Ising model: General setting
Aran Raoufi
Ann. Probab. 48(2): 760-777 (March 2020). DOI: 10.1214/19-AOP1374

Abstract

We prove that at any inverse temperature $\beta $ and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. The theorem is equivalent with the differentiability of the free energy with respect to the temperature at any temperature. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not $\mathbb{Z}^{d}$, or when the graph is $\mathbb{Z}^{d}$ but endowed with infinite-range interactions, or even $\mathbb{Z}^{2}$ with finite-range interactions.

Among the other corollaries of this result, we can list continuity of the magnetization at any noncritical temperature and the uniqueness of FK-Ising infinite-volume measures at any temperature.

Citation

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Aran Raoufi. "Translation-invariant Gibbs states of the Ising model: General setting." Ann. Probab. 48 (2) 760 - 777, March 2020. https://doi.org/10.1214/19-AOP1374

Information

Received: 1 January 2018; Revised: 1 May 2019; Published: March 2020
First available in Project Euclid: 22 April 2020

zbMATH: 07199861
MathSciNet: MR4089494
Digital Object Identifier: 10.1214/19-AOP1374

Subjects:
Primary: 60K35, 82B20

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 2 • March 2020
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