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One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point

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Abstract

We consider the phase separation problem for the one-dimensional ferromagnetic Ising model with long-range two-body interaction, J(n) = n −2+α, where \({n\in {\rm {I\!N}}}\) denotes the distance of the two spins and \({\alpha \in [0,\alpha_+[}\) with α + = (log 3)/(log 2) −1. We prove that when α = 0 the localization of the phase separation fluctuates macroscopically with a non-uniform explicit limiting law, while when 0 < α < α + the macroscopic fluctuations disappear and mesoscopic ones appear with a gaussian behavior when conveniently scaled. The mean magnetization profile is also given.

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Authors and Affiliations

  1. Gran Sasso Science Institute, INFN Center for Advanced Studies, Viale F. Crispi 7, 67100, L’Aquila, Italy

    Marzio Cassandro

  2. Dipartimento di Matematica, Università di L’Aquila, Via Vetoio 1, 67010, Coppito (AQ), Italy

    Immacolata Merola

  3. LATP, CMI, UMR 6632, CNRS, Université de Provence, 39 rue Frederic Joliot Curie, 13453, Marseille Cedex 13, France

    Pierre Picco

  4. Institute of Mathematics, 29 Do’rmon Yo’li str, 100125, Tashkent, Uzbekistan

    Utkir Rozikov

Authors
  1. Marzio Cassandro
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  2. Immacolata Merola
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  3. Pierre Picco
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  4. Utkir Rozikov
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Corresponding author

Correspondence to Pierre Picco.

Additional information

Communicated by F. Toninelli

This work was supported by CNRS-INdAM GDRE 224 GREFI-MEFI, M.C and I.M were supported by Prin07: 20078XYHYS.

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Cassandro, M., Merola, I., Picco, P. et al. One-Dimensional Ising Models with Long Range Interactions: Cluster Expansion, Phase-Separating Point. Commun. Math. Phys. 327, 951–991 (2014). https://doi.org/10.1007/s00220-014-1957-5

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