Replica analysis of the one-dimensional KPZ equation

Summary

In the last few years several exact solutions have been obtained for the onedimensional KPZ equation, which describes the dynamics of growing interfaces. In particular the computations based on replica method have allowed to study fine fluctuation properties of the interface for various initial conditions including the narrow wedge, flat and stationary cases. In addition, an interesting aspect of the replica analysis of the KPZ equation is that the calculations are not only exact but also “almost rigorous”. In this article we give a short review of this development.

1. Introduction

The one-dimensional Kardar–Parisi–Zhang (KPZ) equation,

is a well known prototypical equation which describes a growing interface [Kardar et al. 1986; Barabási and Stanley 1995]. Here h(x, t) represents the height of the surface at position x ∈ ℝ and time t ≥ 0. The first term represents a non linearity effect and the second term describes a smoothing mechanism. The parameters λ and υ measure the strengths of these effects. The last term η(x, t) indicates the existence of randomness in our description of surface growth. For the standard KPZ equation it is taken to be the Gaussian white noise with covariance,

Here and in the remainder of the article, (...) indicates an average with respect to the randomness η.

The KPZ equation (1) is a nonlinear stochastic partial differential equation (SPDE), which is difficult to handle in general. But fortunately the KPZ equation has a nice integrable structure which has allowed detailed studies of its properties. In particular the one-point height distribution has been computed explicitly for three different initial conditions: the narrow wedge h(x,0) = -|x|/δ,δ→0

[Sasamoto and Spohn 2010a; Amir et al. 2011], the flath(x,0)=0,x∈ ℝ [Calabrese and Le Doussal 2011], and the stationary, h(x, 0) = B(x) [Imamura and Sasamoto 2012], cases. Here B(x),x∈ ℝ represents the (two-sided) onedimensional Brownian motion with B(0)= 0. From the narrow wedge initial condition the surface grows to a shape of parabola macroscopically, which is representative of a curved surface. The flat case has been the most typical initial condition for Monte Carlo simulation studies, whereas the stationary case is regarded as one of the most important situation from the point of view of nonequilibrium statistical mechanics