Tightness of discrete Gibbsian line ensembles with exponential interaction Hamiltonians

In this paper we introduce a framework to prove tightness of a sequence of discrete Gibbsian line ensembles $\mathcal{L}^N = \{\mathcal{L}_k^N(x), k \in \mathbb{N}, x \in \frac{1}{N}\mathbb{Z}\}$, which is a collection of countable random curves. The sequence of discrete line ensembles $\mathcal{L}^N$ we consider enjoys a resampling invariance property, which we call $(H^N,H^{RW,N})$-Gibbs property. We also assume that $\mathcal{L}^N$ satisfies technical assumptions A1-A4 on $(H^N,H^{RW,N})$ and the assumption that the lowest labeled curve with a parabolic shift, $\mathcal{L}_1^N(x) + \frac{x^2}{2}$, converges weakly to a stationary process in the topology of uniform convergence on compact sets. Under these assumptions, we prove our main result Theorem 2.18 that $\mathcal{L}^N$ is tight as a line ensemble and that $H$-Brownian Gibbs property holds for all subsequential limit line ensembles with $H(x)= e^x$. As an application of Theorem 2.18, under weak noise scaling, we show that the scaled log-gamma line ensemble $\bar{\mathcal{L}}^N$ is tight, which is a sequence of discrete line ensembles associated with the inverse-gamma polymer model via the geometric RSK correspondence. The $H$-Brownian Gibbs property (with $H(x) = e^x$) of its subsequential limits also follows.