  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.
