Abstract
An interactionU 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 volumesv⊂ℤv, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z v (u+ũ) for the complex perturbationsu+ũ ofu do not vanish for all Vℤv and all Ũ with ∥Ũ∥<ɛ, provided this is true only forv with diam v⩽C(ɛ) and ∥Ũ∥<ɛ′ (but withɛ<ɛ′).
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Dobrushin, R.L., Shlosman, S.B. Completely analytical interactions: Constructive description. J Stat Phys 46, 983–1014 (1987). https://doi.org/10.1007/BF01011153
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DOI: https://doi.org/10.1007/BF01011153