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Variational Principle for Weakly Dependent Random Fields

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Abstract

Using an alternative notion of entropy introduced by Datta, the max-entropy, we present a new simplified framework to study the minimizers of the specific free energy for random fields which are weakly dependent in the sense of Lewis, Pfister, and Sullivan. The framework is then applied to derive the variational principle for the loop O(n) model and the Ising model in a random percolation environment in the nonmagnetic phase, and we explain how to extend the variational principle to similar models. To demonstrate the generality of the framework, we indicate how to naturally fit into it the variational principle for models with an absolutely summable interaction potential, and for the random-cluster model.

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Acknowledgements

The authors are grateful to Nilanjana Datta, Aernout van Enter, Geoffrey Grimmett, James Norris, and Peter Winkler for many useful discussions. The authors would like to express their special gratitude to Nathanaël Berestycki for enabling them to collaborate on this paper. The first author was supported by the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, the UK Engineering and Physical Sciences Research Council Grant EP/L016516/1, and the Shapiro Visitor Program of the Department of Mathematics, Dartmouth College. The second author was supported by the UK Engineering and Physical Sciences Research Council Grant EP/L018896/1.

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Authors and Affiliations

  1. Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, UK

    Piet G. Lammers

  2. Department of Mathematics, Dartmouth College, New Hampshire, USA

    Martin Tassy

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  1. Piet G. Lammers
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  2. Martin Tassy
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Correspondence to Martin Tassy.

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Communicated by Hal Tasaki.

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Lammers, P.G., Tassy, M. Variational Principle for Weakly Dependent Random Fields. J Stat Phys 179, 846–870 (2020). https://doi.org/10.1007/s10955-020-02538-8

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