A replica-coupling approach to disordered pinning models

Abstract

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^{-(\alpha+1)}L(n) with \alpha\ge0 and L(.) slowly varying. We then assign a random, n-dependent reward/penalty to the occurrence of the event that the site n belongs to tau. 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 \tau occupies a finite fraction of N to a delocalized phase where the density of \tau vanishes. In absence of disorder the transition is of first order for \alpha>1 and of higher order for \alpha<1. Moreover, for \alpha ranging from 1 to 0, the transition ranges from first to infinite order. Presence of even an arbitrarily small amount of disorder is known to modify the order of transition as soon as \alpha>1/2. 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 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.