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 ${\mathrm{Gamma}}^{-1}(2\hat{I}¸)$ and diagonal weights as ${\mathrm{Gamma}}^{-1}(\hat{I}\pm +\hat{I}¸)$ for $\hat{I}¸>0$ and $\hat{I}\pm >-\hat{I}¸$. We show that in the bound phase, i.e., when $\hat{I}\pm \in (-\hat{I}¸,0)$, 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.

40 pages, 17 figures; Final version; To appear in Communications in Mathematical Physics