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≥⋯≥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 KMT-type 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 (fN) 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 (fN) 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.