2025/06/09 by Sayan Das, Das, Sayan, Christian Serio +1 · 1 citation
Mathematics · #Stochastic processes and statistical mechanics #Geometry and complex manifolds #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2506.07939
For each α∈ ℝ, t ≥ 1, we show that there exists a unique ℕ-indexed line ensemble of random continuous curves ℝ≤ 0 → 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∈ 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→∞. 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.