Tightness of Bernoulli Gibbsian line ensembles

A Bernoulli Gibbsian line ensemble $\mathfrak{L} = (L_1, \dots, L_N)$ is the law of trajectories $N-1$ independent random walkers $L_1, L_{N-1}$ with possibly initial and terminal locations that are conditioned to never cross each other or a given up-right path $L_N$ (i.e. $L_1 \geq \cdots L_N$). In this paper we investigate asymptotic behavior sequences ensembles $\mathfrak{L}^N (L^N_1, L^N_N)$ when number $N$ tends infinity. We prove if one has mild but uniform control one-point marginals lowest-indexed (or top) curves $L_1^N$ then sequence $\mathfrak{L}^N$ tight in space ensembles. Furthermore, show top converge finite dimensional sense parabolic Airy$_2$ process parabolically shifted Airy ensemble.