Strong Characterization for the Airy Line Ensemble

Abstract

In this paper we show that a Brownian Gibbsian line ensemble whose top curve approximates a parabola must be given by the parabolic Airy line ensemble. More specifically, we prove that if $\mathcal{L}=({\mathcal{L}}_1,{\mathcal{L}}_2,\dots )$ is a line ensemble satisfying the Brownian Gibbs property, such that for any $\epsilon >0$ there exists a constant $\mathfrak{K}(\epsilon )>0$ with $$\mathbb{P}\left[\right|{\mathcal{L}}_1(t)+2^{-1/2}{t}^2|\le \epsilon t^2+\mathfrak{K}(\epsilon )]\ge 1-\epsilon ,\text{for all t∈R},$$ then $\mathcal{L}$ is the parabolic Airy line ensemble, up to an independent affine shift. Specializing this result to the case when $\mathcal{L}(t)+2^{-1/2}{t}^2$ is translation-invariant confirms a prediction of Okounkov and Sheffield from 2006 and Corwin-Hammond from 2014.

263 pages, 45 figures; Version 2: Incorporated referee comments, including reordering and splitting chapters of the paper