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Glassy States: The Free Ising Model on a Tree

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Abstract

We consider the ferromagnetic Ising model on the Cayley tree and we investigate the decomposition of the free state into extremal states below the spin glass temperature. We show that this decomposition has uncountably many components. The tail observable showing that the free state is not extremal is related to the Edwards–Anderson parameter, measuring the variance of the (random) magnetization obtained from drawing boundary conditions from the free state.

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Acknowledgements

Part of this work has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). Part of this work, concerning the EA parameter, has been carried out at IITP RAS. The support of Russian Foundation for Sciences (project No. 14-50-00150) is gratefully acknowledged. This work was partially supported by the CNRS PICS grant “Interfaces aléatoires discrètes et dynamiques de Glauber” and by the grant PRC No. 1556 CNRS-RFBR 2017-2019 ‘Multi-dimensional semi-classical problems of Condensed Matter Physics and Quantum Mechanics ”. CM thanks the hospitality of the CPT-Luminy at Marseille.

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Authors and Affiliations

  1. Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

    Daniel Gandolfo, Jean Ruiz & Senya Shlosman

  2. Instituut voor Theoretische Fysica, KU Leuven, Leuven, Belgium

    Christian Maes

  3. Skolkovo Institute of Science and Technology, Moscow, Russia

    Senya Shlosman

  4. Institute of Information Transmission Problems, RAS, Moscow, Russia

    Senya Shlosman

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  1. Daniel Gandolfo
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  2. Christian Maes
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  3. Jean Ruiz
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  4. Senya Shlosman
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Correspondence to Senya Shlosman.

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Communicated by Ivan Corwin.

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Gandolfo, D., Maes, C., Ruiz, J. et al. Glassy States: The Free Ising Model on a Tree. J Stat Phys 180, 227–237 (2020). https://doi.org/10.1007/s10955-019-02382-5

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