The half-space KPZ line ensemble and its scaling limit
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| Format: | Preprint |
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2025
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| author | Das, Sayan Serio, Christian |
| author_facet | Das, Sayan Serio, Christian |
| contents | For each $α\in \mathbb{R}$, $t \geq 1$, 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 $α$.
(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 $α=μt^{-1/3}$ with $μ\in \mathbb{R}$ fixed (critical regime) and for $α>0$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_07939 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The half-space KPZ line ensemble and its scaling limit Das, Sayan Serio, Christian Probability Mathematical Physics 82D60, 60K35, 60H15, 60J65 For each $α\in \mathbb{R}$, $t \geq 1$, 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 $α$. (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 $α=μt^{-1/3}$ with $μ\in \mathbb{R}$ fixed (critical regime) and for $α>0$ 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. |
| title | The half-space KPZ line ensemble and its scaling limit |
| topic | Probability Mathematical Physics 82D60, 60K35, 60H15, 60J65 |
| url | https://arxiv.org/abs/2506.07939 |