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Entropic Repulsion and Lack of the g-Measure Property for Dyson Models

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Abstract

We consider Dyson models, Ising models with slow polynomial decay, at low temperature and show that its Gibbs measures deep in the phase transition region are not g-measures. The main ingredient in the proof is the occurrence of an entropic repulsion effect, which follows from the mesoscopic stability of a (single-point) interface for these long-range models in the phase transition region.

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Acknowledgements

We thank the referees for a number of helpful remarks and suggestions. We also thank S. Bethuelsen, M. Cassandro, L. Cioletti, D. Conache, R. Fernández, S. Gallo, G. Iacobelli, G. Maillard, F. Paccaut, P. Picco, and E. Verbitskiy for various helpful conversations over the years. We thank Evgeny Verbitskiy for providing us with [4] and Jorge Littin for providing us with [59]. RB is partially supported by the Dutch stochastics cluster STAR, by FAPESP Grants 2011/16265-8, 2016/08518-7 and 2016/25053-8, CNPq Grants 453985/2016-5, 312112/2015-7 and 446658/2014-6. EOE is supported by FAPESP Grants 2014/10637-9 and 2015/14434-8. ALN has benefited from Dutch supports (STAR, EURANDOM, Lorentz Center, TU Delft, RU Groningen) for short research visits to the Netherlands and from Franco-Dutch supports (CNRS, Networks) for a longer research visit at the CNRS UMI Eurandom during the academic year 2017–2018, when this work was achieved.

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Authors and Affiliations

  1. Institute of Mathematics and Statistics - IME, USP - University of São Paulo, São Paulo, Brazil

    Rodrigo Bissacot & Eric O. Endo

  2. Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands

    Eric O. Endo & Aernout C. D. van Enter

  3. Laboratoire de Mathématiques et d’Analyse Appliquées, LAMA UMR CNRS 8050 – Université Paris-Est (UPEC), Créteil, France

    Arnaud Le Ny

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  1. Rodrigo Bissacot
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  2. Eric O. Endo
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Correspondence to Arnaud Le Ny.

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Communicated by H. Duminil-Copin

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Bissacot, R., Endo, E.O., van Enter, A.C.D. et al. Entropic Repulsion and Lack of the g-Measure Property for Dyson Models. Commun. Math. Phys. 363, 767–788 (2018). https://doi.org/10.1007/s00220-018-3233-6

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