We consider the log-gamma polymer in the half-space with bulk weights distributed as \({\text {Gamma}}^{-1}(2\theta )\) and diagonal weights as \({\text {Gamma}}^{-1}(\alpha +\theta )\) for \(\theta >0\) and \(\alpha >-\theta \) . We show that in the bound phase, i.e., when \(\alpha \in (-\theta ,0)\) , the endpoint of the polymer lies within an O(1) stochastic window of the diagonal. This result gives the first rigorous proof of the pinned phenomena for the half-space polymers in the bound phase conjectured by Kardar Kardar (Phys Rev Lett 55:2235, 1985). We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.
