Completely analytical interactions: Constructive description

R. L. Dobrushin, S. B. Shlosman

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106 Citations (Scopus)


An interaction U is called a completely analytical (CA) interaction, if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions Zv(u), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln Zv(u), v→∞. The 12 conditions are presented, together with part of the proof of their equivalence. The main result of the paper is that each condition is constructive in the following sense: instead of checking it in all finite volumes v⊂ℤv, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Zv(u+ũ) for the complex perturbations u+ũ of u do not vanish for all Vℤv and all Ũ with ∥Ũ∥<e{open}, provided this is true only for v with diam v≤C(e{open}) and ∥Ũ∥<e{open}′ (but with e{open}<e{open}′).

Original languageEnglish
Pages (from-to)983-1014
Number of pages32
JournalJournal of Statistical Physics
Issue number5-6
Publication statusPublished - Mar 1987
Externally publishedYes


  • Analyticity
  • correlation decay
  • Gibbs states
  • surgery method
  • uniqueness


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