Tightness and local fluctuation estimates for the KPZ line ensemble.

In this paper we study the KPZ line ensemble $\mathcal{H}^t=\{\mathcal{H}^t_n\}_{n\in\mathbb{N}}$ under the $t^{1/3}$ vertical and $t^{2/3}$ horizontal scaling. We prove quantitative (uniformly in $t$) local fluctuation estimates on curves in $\mathcal{H}^t$ in Theorem 1.3, which enables us to show the tightness of the scaled KPZ line ensembles as $t$ varies (Theorem 1.4(i)). Furthermore, as $t$ increases, the curves in the scaled KPZ line ensemble become more and more ordered. The limit is non-intersecting and enjoys the Brownian Gibbs property, see Theorem 1.4(ii). Together with the recent results in [QS20], [Vir], and [DM], the KPZ line ensemble converges to the Airy line ensemble. We will employ the current estimates in a forthcoming paper [Wu21] to study the Brownian regularity for the scaled KPZ line ensemble.