The periodic Airy line ensembles

In this paper we construct a one-parameter family of periodic Airy line ensembles ${\mathcal{A}}^{\mathrm{p}\mathrm{e}\mathrm{r},R}$ for $R>0$ as weak scaling limits of Borodin’s periodic Schur processes. We establish several properties of ${\mathcal{A}}^{\mathrm{p}\mathrm{e}\mathrm{r},R}$, including a determinantal point process structure with an explicit correlation kernel. We also show that, after a parabolic shift and normalization, ${\mathcal{A}}^{\mathrm{p}\mathrm{e}\mathrm{r},R}$ satisfies, on its domain $[-R\mathrm{/}2,R\mathrm{/}2]$, the same Brownian Gibbs property enjoyed by the usual Airy line ensemble. A main technical contribution of the paper is a general tightness criterion for determinantal line ensembles, which extends the argument used by Johansson in [Comm. Math. Phys. 242 (2003), 277-329] to prove a functional limit theorem for the discrete polynuclear growth model.