Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists of $N$ independent random walks on the integers conditioned not to cross one another, i.e., $L_1 \geq \cdots L_N$. In this paper we provide sufficient conditions for convergence a sequence suitably scaled ensembles $f^N (f_1^N,\dots,f_N^N)$ as number curves tends infinity. Assuming log-concavity and KMT-type coupling walk jump distribution, prove that under mild control one-point marginals top with global parabolic shift, full $(f^N)$ is tight in topology uniform over compact sets, moreover any weak subsequential limit possesses Brownian Gibbs property. If addition converge finite-dimensional distributions $\mathrm{Airy}_2$ process, then result arXiv:2002.00684 implies converges parabolically shifted Airy ensemble. These results apply broad class distributions, including geometric well log-concave distribution whose support forms integer interval.