Bulk heights of the KPZ line ensemble

For $t > 0$, let $\{\mathcal{H}^{(t)}_n, n \in \mathbb{N}\}$ be the KPZ line ensemble with parameter $t$, satisfying the homogeneous $\mathbf{H}$-Brownian Gibbs property with $\mathbf{H}(x) =e^x$. We prove quantitative concentration estimates for the $n$th line $\mathcal{H}^{(t)}_n$ which yield the asymptotics $\mathcal{H}^{(t)}_n = n \log n + o(n^{3/4 + ε})$ as $n \to \infty$. A key step in the proof is a general integration by parts formula for $\mathbf{H}$-Brownian Gibbs line ensembles which yields the identity $\mathbb{E} \exp(\mathcal{H}^{(t)}_{n + 1}(x) - \mathcal{H}^{(t)}_n (x)) = n t^{-1}$ for any $n, t, x$.