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Completely analytical Gibbs fields. (English) Zbl 0569.46043
Statistical physics and dynamical systems. Rigorous results, Pap. 2nd Colloq. Workshop, Köszeg/Hung. 1984, Prog. Phys. 10, 371-403 (1985).

[For the entire collection see Zbl 0557.00018.]

The set of completely analytical interactions is defined by thirteen (!) properties (in the revised version of the article). The striking point here is the fact that all of them are equivalent - and that alone shows the class to be natural. The corresponding Gibbs field is unique and possesses all properties of high temperature fields: exponential decay of correlations, analyticity of free energy, local limit theorem and so on. The proofs of these properties to hold is much easier for CA interactions compared with the usual approach based on cluster expansion, and it seems that CA interactions usually include those for which cluster expansion converges.

The above conditions are of four types. We give below some examples. The interaction $U\in CA$ if:

1. The partition function $Z\left(\stackrel{˜}{U};V,\overline{\sigma }\right)\ne 0$ for all complex perturbation $\stackrel{˜}{U}$ of U such that $\parallel U-\stackrel{˜}{U}\parallel <ϵ$ for some $ϵ>0$, uniformly in volume V and boundary condition $\overline{\sigma }$; or

2. The conditional Gibbs distribution in V, being projected into the subvolume $W\subset V$, depends exponentially week on the values of the corresponding boundary condition $\overline{\sigma }$ in points $t\in \partial V$, which are far from W; or

3. The semiinvariants of conditional Gibbs distribution in V decay exponentially, with the estimate uniform in V, $\overline{\sigma }$; or

4. The logarithm of the partition function $Z\left(V,\overline{\sigma }\right)$ has the asymptotic decomposition into volume and boundary terms -

-PLUS the following restriction: U can be connected with zero interaction by a path U(t), $0\le t\le 1$, such that the same condition holds for all interactions U(t) as well. (For zero interaction they are easy.)

##### MSC:
 46N99 Miscellaneous applications functional analysis 82B05 Classical equilibrium statistical mechanics (general)