The half-space KPZ line ensemble and its scaling limit

TL;DR

This paper constructs and analyzes the half-space KPZ line ensemble, demonstrating its tightness and limiting behavior under KPZ scaling, and introduces new boundary-interacting Brownian motion structures.

Contribution

It introduces the half-space KPZ line ensemble with resampling invariance and boundary interactions, extending understanding of KPZ universality in half-space geometries.

Findings

The line ensemble is tight under KPZ scaling as time goes to infinity.

In the critical regime, the limit matches the half-space Airy line ensemble.

In the supercritical regime, a new structure with pinned Brownian motions emerges.

Abstract

For each $\alpha \in \mathbb{R}$, $t \geq 1$, we show that there exists a unique $\mathbb{N}$-indexed line ensemble of random continuous curves $\mathbb{R}_{\le 0} \to \mathbb{R}$ with the following properties: (1) The top curve is distributed as the time-$t$ Cole--Hopf solution to the half-space KPZ equation with narrow wedge initial condition and Neumann boundary condition with parameter $\alpha$. (2) The line ensemble satisfies a one-sided resampling invariance property, involving softly non-intersecting Brownian motions with an attractive potential between pairs at the boundary. We call this object the half-space KPZ line ensemble. For $\alpha=\mu t^{-1/3}$ with $\mu \in \mathbb{R}$ fixed (critical regime) and for $\alpha>0$ fixed (supercritical regime), we show that the half-space KPZ line ensemble is tight under 1:2:3 KPZ scaling as $t\to\infty$. Moreover, all subsequential limits approximate a parabola and enjoy a one-sided Brownian Gibbs property, described by non-intersecting Brownian motions with pairwise interaction at the boundary. In the critical case this agrees with the half-space Airy line ensemble recently constructed by Dimitrov and Yang. In the supercritical case, we demonstrate a novel structure involving pairwise pinned Brownian motions, one of the main technical contributions of this paper.