Abstract
We study the solution of the Kardar–Parisi–Zhang (KPZ) equation for the stochastic growth of an interface of height h(x, t) on the positive half line, equivalently the free energy of the continuum directed polymer in a half space with a wall at x=0. The boundary condition partial _x h(x,t)|_{x=0}=A corresponds to an attractive wall for A<0, and leads to the binding of the polymer to the wall below the critical value A=-1/2. Here we choose the initial condition h(x, 0) to be a Brownian motion in x>0 with drift -(B+1/2). When A+B rightarrow -1, the solution is stationary, i.e. h(cdot ,t) remains at all times a Brownian motion with the same drift, up to a global height shift h(0, t). We show that the distribution of this height shift is invariant under the exchange of parameters A and B. For any A,B > - 1/2, we provide an exact formula characterizing the distribution of h(0, t) at any time t, using two methods: the replica Bethe ansatz and a discretization called the log-gamma polymer, for which moment formulae were obtained. We analyze its large time asymptotics for various ranges of parameters A, B. In particular, when (A, B) rightarrow (-1/2, -1/2), the critical stationary case, the fluctuations of the interface are governed by a universal distribution akin to the Baik–Rains distribution arising in stationary growth on the full-line. It can be expressed in terms of a simple Fredholm determinant, or equivalently in terms of the Painlevé II transcendent. This provides an analog for the KPZ equation, of some of the results recently obtained by Betea–Ferrari–Occelli in the context of stationary half-space last-passage-percolation. From universality, we expect that limiting distributions found in both models can be shown to coincide.
Highlights
The Kardar–Parisi–Zhang equation [1] describes the stochastic dynamics of the height field, h(x, t), of a growing interface in the continuum, as a function of time, or equivalently, of the free energy of a continuum directed polymer in a random potential as a function of its length
These results have been obtained on the full line, x ∈ R, and have allowed proving the existing conjectures for the scaling exponents of the height fluctuations, δh ∼ t1/3 ∼ x1/2, and to predict and classify the universal probability distributions which arise for various initial conditions
For the TASEP in a half-space, equivalent to last passage percolation in a half-quadrant [3], similar results were obtained in [44,45] using Pfaffian-Schur processes [46], in particular concerning the crossover as the parameter controlling the boundary is varied simultaneously with the distance to the boundary
Summary
The Kardar–Parisi–Zhang equation [1] describes the stochastic dynamics of the height field, h(x, t), of a growing interface in the continuum, as a function of time, or equivalently, of the free energy of a continuum directed polymer in a random potential as a function of its length. For the KPZ equation in the half-space, a solution for the one-point height distribution near the wall valid at any time, was obtained for A = +∞, for the droplet initial condition using the replica Bethe ansatz in [53] (see [54]). Another solution ( non-rigorous) was obtained for A = 0 in [55], with related but different methods using nested contour integral representations. From it we obtain various limits, including our formula for the critical stationary distribution
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