Communications in Mathematical Physics

, Volume 86, Issue 3, pp 327–336 | Cite as

More surprises in the general theory of lattice systems

  • Alan D. Sokal


I use Israel's methods to prove new theorems of “ubiquitous pathology” for classical and quantum lattice systems. The main result is the following: Let Φ be any interaction and ϱ be any translation-invariant equilibrium state for Φ (extremal or not). Then there exists a sequence {Φ k } of interactions converging to Φ, having extremal (or even unique) translation-invariant equilibrium states ϱ k , such that {ϱ k } converges to ϱ. In certain situations the perturbations Φ k −Φ can be chosen to lie in a cone of “antiferromagnetic pair interactions.” I discuss the connection with results of Daniëls and van Enter, and point out an application to the one-dimensional ferromagnetic Ising model with 1/r2 interaction (Thouless effect).


Neural Network Statistical Physic Equilibrium State Complex System General Theory 
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Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Alan D. Sokal
    • 1
  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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