The half-space KPZ line ensemble and its scaling limit

Abstract

For each $\alpha \in \mathbb{R},t\ge 1$, 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 near the boundary 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 supercritical case, one of the main technical contributions of this paper is the first rigorous demonstration of a structure involving pairwise pinned Brownian motions, previously conjectured in [BCD23]. Away from the boundary, we show that the half-space KPZ line ensemble converges to the (full-space) Airy line ensemble under 1:2:3 KPZ scaling as $t\to \infty$. A key ingredient in our arguments, which may be of independent interest, is a novel identity relating the full- and half-space KPZ equations, which may be valuable in deducing various geometric properties of the latter.

arXiv comment78 pages, 2 figures. Bulk convergence results proven in v2