  Half-space directed polymers in random environments are models of interface
growth in the presence of an attractive hard wall. They arise naturally in the
study of wetting and entropic repulsion phenomena. In 1985, Kardar predicted a
"depinning" phase transition as the attractive force of the wall is weakened.
This phase transition has been rigorously established for integrable models of
half-space last passage percolation, i.e. half-space directed polymers at zero
temperature, in a line of study tracing back to work of Baik--Rains. On the
other hand, for integrable positive temperature models, the first rigorous
proof of this phase transition has only been obtained very recently through a
series of works of Barraquand--Wang, Imamura--Mucciconi--Sasamoto [IMS],
Barraquand--Corwin--Das, and Das--Zhu [DZ] on the half-space log-Gamma polymer.
In this paper we study a broad class of half-space directed polymer models with
minimal assumptions on the random environment. We prove that an attractive
force on the wall strong enough to macroscopically increase the free energy
induces phenomena characteristic of the subcritical "bound phase," namely the
pinning of the polymer to the wall and the diffusive fluctuations and limiting
Gaussianity of the free energy. Our arguments are geometric in nature and allow
us to analyze the positive temperature and zero temperature models
simultaneously. Moreover, given the macroscopic free energy increase proven in
[IMS] for the half-space log-Gamma polymer, our arguments can be used to
reprove the results of [IMS, DZ] on polymer geometry and free energy
fluctuations in the bound phase.
