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

Abstract

We consider the log-Gamma polymer in the half-space with bulk weights distributed as $\operatorname{Gamma}^{-1}(2θ)$ and diagonal weights as $\operatorname{Gamma}^{-1}(α+θ)$ for $θ>0$ and $α>-θ$. We show that in the bound phase, i.e., when $α\in (-θ,0)$, 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.