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Probability Theory and Related Fields

, Volume 153, Issue 1–2, pp 25–44 | Cite as

A finite-volume version of Aizenman–Higuchi theorem for the 2d Ising model

  • Loren Coquille
  • Yvan VelenikEmail author
Article

Abstract

In the late 1970s, in two celebrated papers, Aizenman and Higuchi independently established that all infinite-volume Gibbs measures of the two-dimensional ferromagnetic nearest-neighbor Ising model at inverse temperature \({\beta\geq 0}\) are of the form \({\alpha\mu^{+}_\beta + (1-\alpha)\mu^{-}_\beta}\) , where \({\mu^{+}_\beta}\) and \({\mu^{-}_\beta}\) are the two pure phases and \({0\leq\alpha\leq 1}\) . We present here a new approach to this result, with a number of advantages: (a) We obtain an optimal finite-volume, quantitative analogue (implying the classical claim); (b) the scheme of our proof seems more natural and provides a better picture of the underlying phenomenon; (c) this new approach might be applicable to systems for which the classical method fails.

Keywords

Ising model Gibbs states Translation invariance 

Mathematics Subject Classification (2000)

60K35 82B20 

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GenevaGeneva 4Switzerland

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