Tightness of Bernoulli Gibbsian line ensembles

A Bernoulli Gibbsian line ensemble L = (L-1, ... , L-N) is the law of the trajectories of N-1 independent Bernoulli random walkers L-1, ... , LN-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 >= center dot center dot center dot >= L-N). In this paper we investigate the asymptotic behavior of sequences of Bernoulli Gibbsian line ensembles L-N = (L-1(N), ... , 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 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 L-N converge to the parabolic Airy line ensemble.