On the critical behavior of the magnetization in high-dimensional Ising models

M. Aizenman, R. Fernández

Research output: Contribution to journalArticlepeer-review

90 Scopus citations


We derive rigorously general results on the critical behavior of the magnetization in Ising models, as a function of the temperature and the external field. For the nearest-neighbor models it is shown that in d≥4 dimensions the magnetization is continuous at Tc and its critical exponents take the classical values δ=3 and β=1/2, with possible logarithmic corrections at d=4. The continuity, and other explicit bounds, formally extend to d>3 1/2. Other systems to which the results apply include long-range models in d=1 dimension, with 1/|x-y|λ couplings, for which 2/(λ-1) replaces d in the above summary. The results are obtained by means of differential inequalities derived here using the random current representation, which is discussed in detail for the case of a nonvanishing magnetic field.

Original languageEnglish (US)
Pages (from-to)393-454
Number of pages62
JournalJournal of Statistical Physics
Issue number3-4
StatePublished - Aug 1986
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


  • Critical exponents
  • Ising model
  • random-current representation
  • spontaneous magnetization
  • upper critical dimension


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