Persistence of Randomly Coupled Fluctuating Interfaces

Satya N. Majumdar 1, Dibyendu Das 2

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 71 (2005) 036129

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.

  • 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS),
    CNRS : UMR8626 – Université Paris XI - Paris Sud
  • 2. Department of Physics,
    Indian Institute of Technology Bombay