  We consider a polymer, with monomer locations modeled by the trajectory of an
underlying Markov chain, in the presence of a potential thatinteracts with the
polymer when it visits a particular site 0. Disorder is introduced by having
the interaction vary from one monomer to another, as a constant $u$ plus i.i.d.
mean-0 randomness. There is a critical value of $u$ above which the polymer is
pinned, placing a positive fraction (called the contact fraction) of its
monomers at 0 with high probability. When the excursions of the underlying
chain have a finite mean but no finite exponential moment, it is known that the
depinning transition (more precisely, the contact fraction) in the
corresponding annealed system is discontinuous. One generally expects the
presence of disorder to smooth transitions, and it was proved by Giacomin and
Toninelli that when the excursion length distribution has power-law tails, the
quenched system has a continuous transition even if the annealed system does
not. We show here that when the underlying chain is transient but the finite
part of the excursion length distribution has exponential tails, then the
depinning transition is discontinuous even in the quenched system, and the
quenched and annealed critical points are strictly different. By contrast, in
the recurrent case, the depinning behavior depends on the subexponential
prefactors on the exponential decay of the excursion length distribution, and
when these prefactors decay with an appropriate power law, the quenched
transition is continuous even though the annealed one is not.
