Completely analytical interactions: constructive description

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/sub v/(U)< or correlation decay, or truncated correlation bounds, or asymptotic behavior of 1n Z/sub v/(U), V → infinity. 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 an all finite volumes V contains Z/sup v/, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z/sub v/(U + U tilde) for the complex perturbations U + U tilde of U do not vanish for all V contains Z/sup v/ and all U tilde with absolute value of U tilde < epsilon, provided this is true only for V with diam V ≤ C (epsilon) and absolute value of U tilde < epsilon' (but with epsilon < epsilon')