The Super-Additive Ergodic Theorem approach

Let us apply Theorem A.13 to our context below (1), (2) and (3) refer to the three assumptions of the theorem. Choose Fj (u ) = logZ and the super-additivity property (2) is an immediate consequence of (4.10). Hypothesis (1) is just the fact that w and hypothesis (3) is (4.8). In order to obtain the statement of Theorem 4.1 let us call the the limit. Since = F0 , then F , is measurable with respect to the a- 

Therefore for the existence of the free energy it is sufficient that du u and that oji G L. Moreover, a sufficient condition for self-averaging is ergodicity, namely that if A is a measurable subset of D such that OAc A then P(A) = 0 or 1. In fact this directly implies that any measurable function on Cl that is invariant under translations is almost surely a constant. 

Theorem 4.6 If to is a stationary ergodic sequence such that oj G, then the limit of the sequence l/N) ogZlf N exists P(do )-a.s. and in L. Moreover the limit is non-random. 

Remark 4.7 The proof in Section 4- can be generalized (well) beyond the IID case. Reconsidering the proof one realizes that the key point is that a sufficient condition is that if F is a measurable and integrable function on then Ann Y =iF 6 loi,. .., lom))/n = E[F(o ]. And we want 

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