Persistence of Randomly Coupled Fluctuating Interfaces

TL;DR

This paper analyzes the persistence properties of two coupled fluctuating interfaces, revealing that the second interface's height process becomes a fractional Brownian motion with a specific Hurst exponent, and confirming the results through simulations.

Contribution

It introduces a model of two coupled interfaces with one influenced by a quenched random velocity field, deriving analytical expressions for persistence exponents and validating them numerically.

Findings

The second interface's height process is a fractional Brownian motion with Hurst exponent H_2=1-eta_1/2.

The persistence exponent for the second interface is -H_2=eta_1/2.

Analytical results are confirmed by numerical simulations.

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\to 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\to \infty, 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.