Tightness of discrete Gibbsian line ensembles

TL;DR

This paper establishes conditions under which discrete Gibbsian line ensembles converge to the Airy line ensemble, extending understanding of their asymptotic behavior and the universality of the Airy process.

Contribution

It provides sufficient conditions for the convergence of scaled discrete Gibbsian line ensembles to the Airy line ensemble, including log-concavity and coupling assumptions.

Findings

Proves tightness of the sequence of line ensembles under mild conditions.

Shows subsequential limits have the Brownian Gibbs property.

Demonstrates convergence to the Airy line ensemble under top curve convergence.

Abstract

A discrete Gibbsian line ensemble $\mathfrak{L} = (L_1,\dots,L_N)$ consists of $N$ independent random walks on the integers conditioned not to cross one another, i.e., $L_1 \geq \cdots \geq L_N$. In this paper we provide sufficient conditions for convergence of a sequence of suitably scaled discrete Gibbsian line ensembles $f^N = (f_1^N,\dots,f_N^N)$ 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 $(f^N)$ 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 $\mathrm{Airy}_2$ process, then a result of arXiv:2002.00684 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.