Half-space stationary Kardar–Parisi–Zhang equation beyond the Brownian case. (8th July 2022)

• Record Type: Journal Article
• Authors: Barraquand, Guillaume; Krajenbrink, Alexandre; Le Doussal, Pierre
• Is Part Of: Journal of physics. Volume 55:Number 27(2022)
• Publication Date: 2022-07-08
• Subjects: Kardar–Parisi–Zhang equation -- directed polymers -- exactly solvable models -- Fredholm determinants -- Tracy–Widom distribution -- stationary distribution -- KPZ universality class; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/ac761d
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We study the Kardar–Parisi–Zhang (KPZ) equation on the half-line x ⩾ 0 with Neumann type boundary condition. Stationary measures of the KPZ dynamics were characterized in recent work: they depend on two parameters, the boundary parameter u of the dynamics, and the drift − v of the initial condition at infinity. We consider the fluctuations of the height field when the initial condition is given by one of these stationary processes. At large time t, it is natural to rescale parameters as ( u, v ) = t −1/3 ( a, b ) to study the critical region. In the special case a + b = 0, treated in previous works, the stationary process is simply Brownian. However, these Brownian stationary measures are particularly relevant in the bound phase ( a < 0) but not in the unbound phase. For instance, starting from the flat or droplet initial condition, the height field near the boundary converges to the stationary process with a > 0 and b = 0, which is not Brownian. For a + b ⩾ 0, we determine exactly the large time distribution F a, b stat of the height function h (0, t ). As an application, we obtain the exact covariance of the height field in a half-line at two times 1 ≪ t 1 ≪ t 2 starting from stationary initial condition, as well as estimates, when starting from droplet initial condition, in the limit t 1 / t 2 → 1.