Dobrushin, R. L.; Shlosman, S. B. Completely analytical interactions: Constructive description. (English) Zbl 0683.60080 J. Stat. Phys. 46, No. 5-6, 983-1014 (1987). Summary: 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 \(Z_ V(U)\), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln \(Z_ Vv(U)\), \(V\to \infty\). 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 \(V\subset {\mathbb{Z}}^ v\), it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions \(Z_ V(U+\tilde U)\) for the complex perturbations \(U+\tilde U\) of U do not vanish for all \(V\subset {\mathbb{Z}}^ v\) and all \(\tilde U\) with \(\| \tilde U\| <\epsilon\), provided this is true only for V with diam \(V\leq C(\epsilon)\) and \(\| \tilde U\| <\epsilon '\) (but with \(\epsilon <\epsilon ')\). Cited in 1 ReviewCited in 78 Documents MSC: 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B99 Equilibrium statistical mechanics Keywords:correlation decay; Gibbs states; interaction; analyticity properties of the partition functions; perturbations × Cite Format Result Cite Review PDF Full Text: DOI References: [1] R. L. Dobrushin and S. B. Shlosman, Completely analytical Gibbs fields, inStatistical Physics and Dynamical Systems (Birkhäuser, 1985). · Zbl 0569.46043 [2] R. L. Dobrushin and S. B. Shlosman, Preprint (1986). [3] R. L. Dobrushin and S. B. Shlosman, Constructive criterion for the uniqueness of Gibbs field, inStatistical Physics and Dynamical Systems (Birkhäuser, 1985). · Zbl 0569.46042 [4] R. L. Dobrushin, I. Kolafa, and S. B. Shlosman, Phase diagram of the two-dimensional Ising antiferromagnet,Commun. Math. Phys. 102:89-103 (1985). · doi:10.1007/BF01208821 [5] R. L. Dobrushin, Asymptotic behavior of Gibbs fiels of lattice systems as a function of the shape of the volume,Teor. Mat. Fiz. 12:115-134 (1972) (in Russian). [6] F. K. Abdulla-Zadeh, R. A. Minlos, and S. K. Pogosyan, Cluster estimates for Gibbs random fields and some applications, inMulticomponent random systems (Marcel Dekker, 1980), pp. 1-36; S. K. Pogosyan,Commun. Math. Phys. 95:227 (1984). [7] R. L. Dobrushin, S. B. Shlosman, The problem of translation invariance in statistical mechanics. Sov. Math. Rev., ser C, v. 5, 1983. · Zbl 0613.76010 [8] R. L. Dobrushin, The prescribtion of the systems of random variables by help of conditional distributions. Teor. ver. prim., 15, N 3, 469-479, 1970. [9] R. L. Dobrushin and E. A. Percherski, Uniqueness conditions for finitely dependent random fields. In: ?Random fields?, v. 1, North-Holland, Amsterdam-Oxford-N.Y., 223-262, 1981. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.