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We consider a renewal process \tau={\tau_0,\tau_1,...} on the integers, wherethe law of \tau_i-\tau_{i-1} has a power-like tailP(\tau_i-\tau_{i-1}=n)=n^{-(\alpha+1)}L(n) with \alpha\ge0 and L(.) slowlyvarying. We then assign a random, n-dependent reward/penalty to the occurrenceof 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 aone-dimensional random defect, (1+1)-dimensional models of wetting ofdisordered substrates, and the Poland-Scheraga model of DNA denaturation. Byvarying the average of the reward, the system undergoes a transition from alocalized phase where \tau occupies a finite fraction of N to a delocalizedphase where the density of \tau vanishes. In absence of disorder the transitionis 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 theorder of transition as soon as \alpha>1/2. In physical terms, disorder isrelevant 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. Alexanderthat, if disorder is sufficiently weak, critical exponents are not modified byrandomness: disorder is irrelevant. In this work, generalizing techniques whichin the framework of spin glasses are known as replica coupling andinterpolation, 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 whichimproves the annealed one.Comment: 12 pages; v2: added Theorem 2.6, typos correcte

A Replica-Coupling Approach to Disordered Pinning Models