In this paper, we establish the ergodicity of the Airy line ensemble with respect to horizontal shifts. This shows that it is the only candidate for Conjecture 3.2 in Corwin & Hammond, Invent. Math. 2014, regarding the classification of ergodic line ensembles satisfying a certain Brownian Gibbs property after a parabolic shift.
"Ergodicity of the Airy line ensemble." Electron. Commun. Probab. 19 1 - 11, 2014. https://doi.org/10.1214/ECP.v19-3504