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Phase Transition for the Ising Model on the Critical Lorentzian Triangulation

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Abstract

The ferromagnetic Ising model without external field on an infinite Lorentzian triangulation sampled from the uniform distribution is considered. We prove uniqueness of the Gibbs measure in the high temperature region and coexistence of at least two Gibbs measures at low temperature. The proofs are based on the disagreement percolation method and on a variant of the Peierls contour method. The critical temperature is shown to be constant a.s.

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Notes

  1. We require that the embedding maps v 0 to the root vertex of T, and sends horizontal/vertical edges of Γ to edges of the same type in T.

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Acknowledgements

The work of M.K. is partially supported by Projet du conseil scientifique BQR 2007, Unité Mixte de Recherche UMR7502, IAEM 0039. A.Y. thanks National Council for Scientific and Technological Development (CNPq), Brazil, grants “Rede Matemática Brasil–França” (306092/2007-7) and “Edital Universal 2006” (471925/2006-3).

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  1. Maxim Krikun

    Present address: Google Ireland, Dublin, Ireland

Authors and Affiliations

  1. Institut Élie Cartan Nancy (IECN), CNRS, INRIA, Nancy-Université, Boulevard des Aiguillettes, B.P. 239, 54506, Vandœuvre-lès-Nancy, France

    Maxim Krikun

  2. Department of Statistics, Institute of Mathematics and Statistics, University of São Paulo, Rua do Matão 1010, 05508-090, São Paulo, SP, Brazil

    Anatoly Yambartsev

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  1. Maxim Krikun
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Correspondence to Anatoly Yambartsev.

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Krikun, M., Yambartsev, A. Phase Transition for the Ising Model on the Critical Lorentzian Triangulation. J Stat Phys 148, 422–439 (2012). https://doi.org/10.1007/s10955-012-0548-0

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