Tightness of Bernoulli Gibbsian line ensembles

Abstract

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