Communications in Mathematical Physics

, Volume 334, Issue 2, pp 719–742 | Cite as

Random Currents and Continuity of Ising Model’s Spontaneous Magnetization

  • Michael AizenmanEmail author
  • Hugo Duminil-Copin
  • Vladas Sidoravicius


The spontaneous magnetization is proved to vanish continuously at the critical temperature for a class of ferromagnetic Ising spin systems which includes the nearest neighbor ferromagnetic Ising spin model on \({\mathbb{Z}^d}\) in d = 3 dimensions. The analysis also applies to higher dimensions, for which the result is already known, and to systems with interactions of power law decay. The proof employs in an essential way an extension of the Ising model’s random current representation to the model’s infinite volume limit. Using it, we relate the continuity of the magnetization to the vanishing of the free boundary condition Gibbs state’s long range order parameter. For reflection positive models the resulting criterion for continuity may be established through the infrared bound for all but the borderline lower dimensional cases. The exclusion applies to the one dimensional model with 1/r 2 interaction for which the spontaneous magnetization is known to be discontinuous at T c .


Ising Model Spontaneous Magnetization Free Boundary Condition Gibbs State Range Order Parameter 
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Copyright information

© The Author(s) 2014

Authors and Affiliations

  • Michael Aizenman
    • 1
    Email author
  • Hugo Duminil-Copin
    • 2
  • Vladas Sidoravicius
    • 3
  1. 1.Departments of Physics and MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Département de MathématiquesUniversité de GenèveGenevaSwitzerland
  3. 3.IMPARio de JaneiroBrazil

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