  Recent results have lead to substantial progress in understanding the role of
disorder in the (de)localization transition of polymer pinning models. Notably,
there is an understanding of the crucial issue of disorder relevance and
irrelevance that is now rigorous. In this work, we exploit interpolation and
replica coupling methods to obtain sharper results on the irrelevant disorder
regime of pinning models. In particular, in this regime, we compute the first
order term in the expansion of the free energy close to criticality and this
term coincides with the first order of the formal expansion obtained by field
theory methods. We also show that the quenched and quenched averaged
correlation length exponents coincide, while, in general, they are expected to
be different. Interpolation and replica coupling methods in this class of
models naturally lead to studying the behavior of the intersection of certain
renewal sequences and one of the main tools in this work is precisely renewal
theory and the study of these intersection renewals.
