On canonical Gibbs states, symmetric and tail events

Abstract

The identification of extremal canonical Gibbs states as Gibbs states is a problem that arises as a natural generalization of de Finetti's theorem on exchangeable 0-1 variables to the non-independent case. If the underlying space carries a shift structure, an affirmative answer has been given for the shift invariant states by Thompson [8] and the author [4]. In the absence of any shift structure, Logan [6] has solved the same problem by investigating the reversible states for certain Markov processes describing the jumps of interacting particles. T. Shiga (private communication) claims to be able to prove the converse (the identification of Gibbs states as extremal canonical Gibbs states) by "dynamical" methods, too. The aim of this article is to give a proof of these and related results by "pure equilibrium" techniques. Our conditions are essentially those of Logan [6], with the difference that we are not restricted to finite range interactions with the condition [6, (5.5)]. Section 1 contains the definition of Gibbs states and canonical Gibbs states. The key result is the existence of a tail measurable "activity indicating" function (Proposition (2.3)) which will be proved in Section 4. From this we deduce in Section 2 the answer to the problem mentioned above (Theorem (2.4)). In particular, we are able to characterize the Gibbs states with fixed activity in the class of all canonical Gibbs states (Theorem (2.6)). As an application we obtain the solution of the introductory problem in the lattice case for the shift invariant states, too (Corollary (2.7)). In Section 3 we prove that for all canonical Gibbs states the tail field and the a-field of all symmetric events almost surely coincidea result which is well-known for symmetric states in the context of the HewittSavage 0-1 law. In the final Section 5 we discuss the question how close our conditions on the interaction are to necessary ones.