A replica-coupling approach to disordered pinning models

We consider a renewal process tau = {tau(0), tau(1),...} on the integers, where the law of tau i - tau(i-1) has a power-like tail P(tau(i) - tau(i-1) = n) = n(-(a+1)) L(n) with alpha >= 0 and L(center dot) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to t. In such generality this class of problems includes, among others, (1+ d)-dimensional models of pinning of directed polymers on a one-dimensional random defect, (1 + 1)-dimensional models of wetting of disordered substrates, and the Poland-Scheraga model of DNA denaturation. By varying the average of the reward, the system undergoes a transition from a localized phase, where t occupies a finite fraction of N to a delocalized phase, where the density of t vanishes. In absence of disorder (i.e., if the reward is independent of n), the transition is of first order for alpha > 1 and of higher order for alpha 1. Moreover, for a ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small (but extensive) amount of disorder is known to modify the order of transition as soon as alpha > 1/2 [11]. In physical terms, disorder is relevant in this situation, in agreement with the heuristic Harris criterion. On the other hand, for 0 < alpha < 1/2 it has been proven recently by K. Alexander [2] that, if disorder is sufficiently weak, critical exponents are not modified by randomness: disorder is irrelevant. In this work, generalizing techniques which in the framework of spin glasses are known as replica coupling and interpolation, we give a new, simpler proof of the main results of [2]. Moreover, we (partially) justify a small-disorder expansion worked out in [9] for alpha < 1/2, showing that it provides a free energy upper bound which improves the annealed one.