Existence of the Free Energy and Self-Averaging

This section is devoted to the proof of the existence of the quenched free energy. The free energy of a disordered system is a priori a random variable, however we will show that it is a degenerate random variable, i.e. a constant this phenomenon is called self-averaging property of the free energy. 

The first step in the proof is controlling the so-caUed quenched averaged free energy by the bound on Hn,u(t) in point (1) in Section 4.1, and by the hypothesis on cu, logZ L. As a matter of fact the same bound implies directly 

So that the sequence with which we wish to define, in the limit, the quenched averaged free energy is bounded above. By using the same esti- 

By taking the P-expectation, keeping in mind that logZ log coincide in law, we obtain the statement. 

Combining Proposition 4.2 and Proposition A.12 the existence of the limit of the sequence ElogZ is established and we set 

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