An Improved Lower Bound on the Free Energy

The result we want to prove is that a quenched sequence of charges, even if they are centered and so on the average the charge is zero, plays successfully in favor of localization (the analogous result for weakly inhomogeneous models is proven in Proposition 3.8). This is of course due to the fact that the typical polymer trajectories target positive charges. We introduce the 

Remark 5.3 Notice that the results in Theorem 5.2(2) and Theorem 2.1, applied to the annealed model (i.e. with j3 replaced by ogbA (3)), match, up to a multiplicative constant, for (3 small. 

The key idea for proving Theorem 5.2 is the following lemma, in which we rewrite as P-expectation of a Boltzmaim factor in which the energy 

We need the following notation given t, for any n odd natural number 

Proof of Theorem 5.2. We assume without loss of generality that Tk = 1 (Remark 1.19) and A 0. Fix u and apply (A.10) with X = Xu t) equal to the expression in the exponent in the right-hand side of (5.12). The measure p is the law of r when the inter-arrival distribution is Kb -), A 0. We are therefore using the strategy of homogeneous localization (Appendix B.l) if rtiK 00 there is no need of localizing and one can take b = 0. 

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