The Strongly Delocalized Regime

Stronger delocalization results may be obtained deeply inside I . By deeply inside T we mean in 

By Jensen inequality Vs C V. The subscript s stands for strongly delocalized and the definition is natmally generalized to copolymers and copolymers with adsorption. Note that (/ , h) Vs if and only if /i logS/c + logM(/ ). 

Theorem 8.6 If (/ , h) Vs then there exists c 0 such that for every N 

Considering still the constrained case, call Em, rt2 the event in the left-hand side of (8.29). The same steps as in (8.30) apply and we obtain 

The proof in the free endpoint case is very similar. Still for (/ , h) GVg, i.e. / 0, we have 

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We give now a scaling limit result in the strongly delocalized regime under the condition that if(-) decays sufficiently rapidly. One is certainly tempted to conjecture the validity of such a result in the whole class of inter-arrival distributions that we consider and everywhere in T>. We will come back to this issue in the next section (with no definite answer). 

In the strongly delocalized regime, from Theorem 8.6 one can extract the weak convergence statements of Corollary 8.7 for general return time distribution, but only along subsequences. 

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