KPZ exponents for the half-space log-gamma polymer

We consider the point-to-point log-gamma polymer of length 2N in a half-space with i.i.d. \({\text {Gamma}}^{-1}(2\theta )\) ${\text{Gamma}}^{-1}(2\theta )$ distributed bulk weights and i.i.d. \({\text {Gamma}}^{-1}(\alpha +\theta )\) ${\text{Gamma}}^{-1}(\alpha +\theta )$ distributed boundary weights for \(\theta >0\) $\theta >0$ and \(\alpha >-\theta \) $\alpha >-\theta$ . We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when \(\alpha =N^{-1/3}\mu \) $\alpha =N^{-1/3}\mu$ for \(\mu \in \mathbb {R}\) $\mu \in \mathbb{R}$ fixed (critical regime) and when \(\alpha >0\) $\alpha >0$ is fixed (supercritical regime). In particular, in these two regimes, we show that after appropriate centering, the free energy process with spatial coordinate scaled by \(N^{2/3}\) $N^{2/3}$ and fluctuations scaled by \(N^{1/3}\) $N^{1/3}$ is tight. These regimes correspond to a polymer measure which is not pinned at the boundary. This is the first instance of establishing the 2/3 transversal exponent for a positive temperature half-space model, and the first instance of the 1/3 fluctuation exponent besides precisely at the boundary where recent work of Imamura et al. (Solvable models in the KPZ class: approach through periodic and free boundary Schur measures. arXiv:2204.08420. 2022) applies and also gives the exact one-point fluctuation distribution (our methods do not access exact fluctuation distributions). Our proof relies on two inputs—the relationship between the half-space log-gamma polymer and half-space Whittaker process (facilitated by the geometric RSK correspondence as initiated in Corwin et al. (Duke Math J 163(3):513–563, 2014), O’Connell et al. (Invent Math 197(2):361–416, 2014), and an identity in Barraquand and Wang (Int Math Res Not 2023:11877, 2022) which relates the point-to-line half-space partition function to the full-space partition function for the log-gamma polymer. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop, in the spirit of work initiated in Corwin and Hammond (Invent Math 195(2):441–508, 2014), a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles.