Half-Space Stationary Kardar-Parisi-Zhang Equation

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 $A+B\to -1$ , the solution is stationary, i.e. $h(·,t)$ 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 $(A,B)\to (-1/2,-1/2)$ , 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.

Keywords: Growth process; Halfspace; Kardar–Parisi–Zhang; Lieb–Liniger; Random matrix theory; Stationary measure.

Fig. 1

Phase diagram indicating the distribution of height fluctuations at large time, as a function of the parameters A, B. The nature of fluctuations in the dashed area around $(A,B)=(-1/2,-1/2)$ is explained in Fig. 2

Fig. 2

Zoom into the vicinity of $(A,B)=(-1/2,-1/2)$. The distribution of height fluctuations at large time is indicated as a function of parameters $a=t^{1/3}(A+\frac{1}{2}),b=t^{1/3}(B+\frac{1}{2})$

Fig. 3

Left: Critical stationary CDF F. Right: corresponding PDF. See Fig. 6 in Appendix. D for the comparison with the asymptotics ($s\to \pm \infty$) in true and logarithmic scales

Fig. 4

An admissible path in the half space log-gamma polymer model, that is a path proceeding by unit steps rightward and upward in the half quadrant $\{(i,j)\in {\mathbb{Z}}_{>0}^2:i⩾j\}$

Fig. 5

The two types of elementary local transformations of down-right paths considered in the proof of Proposition 4.5. The thick black path represents an arbitrary down-right path. The portions in red represent the local modifications of the path that we consider

Fig. 6

Overlap of the left and right tails of the PDF of the critical stationary case (derivative of Eqs. (D.10) and (D.4)) with the complete PDF (derivative of Eq. (7.37)). Top. True scale. Bottom. Logarithmic scale on the vertical axis