Tightness of discrete Gibbsian line ensembles

A discrete Gibbsian line ensemble L = (L1, ... , LN) consists of N independent random walks on the integers conditioned not to cross one another, i.e., L1 > center dot center dot center dot > LN. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles fN = (f1N ,...,fNN) as the number of curves N tends to infinity. Assuming log-concavity and a KMTtype coupling for the random walk jump distribution, we prove that under mild control of the one-point marginals of the top curves with a global parabolic shift, the full sequence ( f N) is tight in the topology of uniform convergence over compact sets, and moreover any weak subsequential limit possesses the Brownian Gibbs property. If in addition the top curves converge in finite-dimensional distributions to the parabolic Airy2 process, then a result of Dimitrov (2021) implies that ( f N) converges to the parabolically shifted Airy line ensemble. These results apply to a broad class of discrete jump distributions, including geometric as well as any log-concave distribution whose support forms a compact integer interval. (c) 2023 Elsevier B.V. All rights reserved.