The phase transition in a general class of Ising-type models is sharp

M. Aizenman, D. J. Barsky, R. Fernández

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108 Scopus citations


For a family of translation-invariant, ferromagnetic, one-component spin systems-which includes Ising and φ{symbol}4 models-we prove that (i) the phase transition is sharp in the sense that at zero magnetic field the high- and low-temperature phases extend up to a common critical point, and (ii) the critical exponent β obeys the mean field bound β≤1/2. The present derivation of these nonperturbative statements is not restricted to "regular" systems, and is based on a new differential inequality whose Ising model version is M≤βhχ+M3+ βM2∂M/∂β. The significance of the inequality was recognized in a recent work on related problems for percolation models, while the inequality itself is related to previous results, by a number of authors, on ferromagnetic and percolation models.

Original languageEnglish (US)
Pages (from-to)343-374
Number of pages32
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - May 1987
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Ising model
  • Phase transition
  • critical exponents
  • inequalities
  • intermediate phase
  • φ{symbol}


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