The half-space log-Gamma polymer in the bound phase

We consider the log-Gamma polymer in the half-space with bulk weights distributed as ${\mathrm{Gamma}}^{-1}(2\theta )$ and diagonal weights as ${\mathrm{Gamma}}^{-1}(\alpha +\theta )$ 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(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.