Preliminary Observations on the Free Energy

We collect in the next proposition a number of soft results on f(-, ) and hc -), which is a function from [0,00) to R. Note in particular that formula (5.3) does not depend on a, the decay exponent of K -). 

Proof First of all note that (5.2) implies (5.3). Moreover the fact that f(-, ) is convex implies that T = (/ , ft-) f(/3, ft) 0 is a convex set. Since clearly (0, h) T for every ft 0, this proves that ftc(-) is convex and the continuity property follows. 

Let us establish the lower bound in (5.2). Remark that the function / 1- E log Zif is convex and that 

For what concerns the upper bound, we just apply the annealed bound procedure, as in (1.72). Therefore we have 

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