Persistence exponents for fluctuating interfaces

Abstract

Numerical and analytic results for the exponent θdescribing the decay of the first return probability of an interface to its initial height are obtained for a large class of linear Langevin equations. The models are parametrized by the dynamic roughness exponent β, with 0 < β< 1; for β= 1/2 the time evolution is Markovian. Using simulations of solid-on-solid models, of the discretized continuum equations as well as of the associated zero-dimensional stationary Gaussian process, we address two problems: The return of an initially flat interface, and the return to an initial state with fully developed steady state roughness. The two problems are shown to be governed by different exponents. For the steady state case we point out the equivalence to fractional Brownian motion, which has a return exponent θ_S = 1 - β. The exponent θ_0 for the flat initial condition appears to be nontrivial. We prove that θ_0 \to \infty for β\to 0, θ_0 \geq θ_S for β< 1/2 and θ_0 \leq θ_S for β> 1/2, and calculate θ_{0,S} perturbatively to first order in an expansion around the Markovian case β= 1/2. Using the exact result θ_S = 1 - β, accurate upper and lower bounds on θ_0 can be derived which show, in particular, that θ_0 \geq (1 - β)^2/βfor small β.