Persistence of randomly coupled fluctuating interfaces

Abstract

We study the persistence properties in a simple model of two coupled interfaces characterized by heights h(1) and h(2), respectively, each growing over a d-dimensional substrate. The first interface evolves independently of the second and can correspond to any generic growing interface, e.g., of the Edwards-Wilkinson or of the Kardar-Parisi-Zhang variety. The evolution of h(2), however, is coupled to h(1) via a quenched random velocity field. In the limit d -> 0, our model reduces to the Matheron-de Marsily model in two dimensions. For d=1, our model describes a Rouse polymer chain in two dimensions advected by a transverse velocity field. We show analytically that after a long waiting time t(0)->infinity, the stochastic process h(2), at a fixed point in space but as a function of time, becomes a fractional Brownian motion with a Hurst exponent, H-2=1-beta(1)/2, where beta(1) is the growth exponent characterizing the first interface. The associated persistence exponent is shown to be theta(s)(2)=1-H-2=beta(1)/2. These analytical results are verified by numerical simulations.