Copolymers and Homogeneous Localization

This section is devoted to applying the homogeneous localization strategy, already exploited in Section 5.2 in the context of disordered pinning models. But let us first prove the soft part of Theorem 6.1, namely  

The apparently strange conditions are simply requiring (1) that the polymer is localized if 0, as long as A 0 (in fact knowing localization for A (0, e), by convexity of f(-, h) and the fact that f(0, 0) = 0, implies localization on all the semi-axis), and (2) that the system is delocalized on a line with sufficiently large slope, at least close to the origin. By monotonicity in h this implies delocalization above this fine. 

The fact that A i— hc X) is non-decreasing is an immediate consequence of the fact that F -,h) is non-decreasing, which in turn follows from the fact that f(0, h) = 0 and from the convexity and non-negativity of f(, h). But we want to prove strict monotonicity and the argument goes as follows. Choose A A and assume hc X ) oo (otherwise there is nothing to prove). By convexity we have Uc X ) 5+Uo(A)(A — A) - - Uc(A), with 5+ the derivative from the right, cf. Appendix A.1.1. Therefore 

Theorem 6.3 For every a 0 there exist two positive constant ci C2 such that 

Both Cl and C2 can be made rather explicit. In fact C2 is very explicit (and independent of a ), see (6.12). On the other hand ci depends on a and an explicit bound on ci can be extracted from (6.17). 

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