Bulk heights of the KPZ line ensemble

For $t>0$, let $\{{\mathcal{H}}_n^{(t)},n\in \mathbb{N}\}$ 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 ${\mathcal{H}}_n^{(t)}$ which yield the asymptotics ${\mathcal{H}}_n^{(t)}=n\log n+o(n^{3\mathrm{/}4+\epsilon })$ as $n\to \mathrm{\infty }$. A key step in the proof is a general integration by parts formula for $\mathbf{H}$-Brownian Gibbs line ensembles which yields the identity $\mathbb{E}\exp ({\mathcal{H}}_{n+1}^{(t)}(x)-{\mathcal{H}}_n^{(t)}(x))=n{t}^{-1}$ for any $n,t,x$.