Half-space polynuclear growth: KPZ universality

Abstract

We study polynuclear growth (PNG) on the half-line with Poisson nucleations in the bulk and at the boundary. For every nonnegative boundary nucleation rate, we construct stationary measures and jointly stationary couplings under a common graphical environment, with profiles pinned at the origin. We realize these couplings as a measurable stationary PNG horizon and prove convergence to the half-space stationary horizon under Kardar-Parisi-Zhang (KPZ) scaling. The horizon convergence includes its base profile and positive representing measure, together with joint profile convergence at every prescribed finite parameter list. We also prove convergence of the height evolution from general upper-semicontinuous initial profiles to the half-space KPZ fixed point. The associated passage field converges locally uniformly to the half-space directed landscape, jointly with evolutions from finitely many initial profiles driven by the same environment. The convergence results cover the critical boundary window and every fixed subcritical boundary rate, including zero. All fixed subcritical rates produce the same limiting laws.