Bulk heights of the KPZ line ensemble

TL;DR

This work analyzes the bulk behavior of the KPZ$_t$ line ensemble with $\mathbf{H}(x)=e^x$ and derives sharp concentration results for the $n$th line, showing $\mathcal{H}^{(t)}_n(0)=e_n^{(t)}+o(n^{3/4+\varepsilon})$ where $e_n^{(t)}=\log(t^{1-n}(n-1)!)+t/24$. A central tool is a general integration-by-parts formula for $\mathbf{H}$-Brownian Gibbs ensembles, yielding $\mathbb{E}\exp(\mathcal{H}^{(t)}_{n+1}-\mathcal{H}^{(t)}_n)=n/t$, which informs the bulk scaling and motivates a Toda-lattice-type energy balance. The authors develop a parabolic Toda boundary problem and use monotone couplings to transfer these insights to the KPZ line ensemble, obtaining detailed tail bounds that imply a strong law of large numbers for $\mathcal{H}^{(t)}_n(0)$ and a deterministic Toda-like spacing in the bulk. The results illuminate how finite-temperature line ensembles can exhibit bulk drift to $+\infty$ with index, and they lay groundwork toward a geometric characterization of the KPZ sheet via connections to the multi-layer stochastic heat equation and the directed landscape.

Abstract

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 $n, t, x$.

Theorems & Definitions (41)

• Theorem 1.1
• Corollary 1.2
• Remark 1.3
• Remark 1.4: Future work
• Theorem 1.5
• Conjecture 1.6
• Conjecture 1.7
• Conjecture 1.8
• Definition 2.1
• Definition 2.2