  In this paper we look at the pinning of a directed polymer by a
one-dimensional linear interface carrying random charges. There are two phases,
localized and delocalized, depending on the inverse temperature and on the
disorder bias. Using quenched and annealed large deviation principles for the
empirical process of words drawn from a random letter sequence according to a
random renewal process [Birkner, Greven and den Hollander, Probab. Theory
Related Fields 148 (2010) 403-456], we derive variational formulas for the
quenched, respectively, annealed critical curve separating the two phases.
These variational formulas are used to obtain a necessary and sufficient
criterion, stated in terms of relative entropies, for the two critical curves
to be different at a given inverse temperature, a property referred to as
relevance of the disorder. This criterion in turn is used to show that the
regimes of relevant and irrelevant disorder are separated by a unique inverse
critical temperature. Subsequently, upper and lower bounds are derived for the
inverse critical temperature, from which sufficient conditions under which it
is strictly positive, respectively, finite are obtained. The former condition
is believed to be necessary as well, a problem that we will address in a
forthcoming paper. Random pinning has been studied extensively in the
literature. The present paper opens up a window with a variational view. Our
variational formulas for the quenched and the annealed critical curve are new
and provide valuable insight into the nature of the phase transition. Our
results on the inverse critical temperature drawn from these variational
formulas are not new, but they offer an alternative approach, that is, flexible
enough to be extended to other models of random polymers with disorder.
