Completely analytical interactions: Constructive description

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 Z_v(u), or correlation decay, or truncated correlation bounds, or asymptotic behavior of ln Z_v(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??^v, it is enough to consider only (a finite amount of) volumes with restricted size. In particular, the partition functions Z _v (u+?) for the complex perturbationsu+? ofu do not vanish for all V? ^v and all ? with ∥?∥<?, provided this is true only forv with diam v?C(?) and ∥?∥<?′ (but with?<?′).