Persistence exponents for fluctuating interfaces
Résumé
Numerical and analytic results for the exponent \\theta describing 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 \\beta, with 0 < \\beta < 1; for \\beta = 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 \\theta_S = 1 - \\beta. The exponent \\theta_0 for the flat initial condition appears to be nontrivial. We prove that \\theta_0 \\to \\infty for \\beta \\to 0, \\theta_0 \\geq \\theta_S for \\beta < 1/2 and \\theta_0 \\leq \\theta_S for \\beta > 1/2, and calculate \\theta_{0,S} perturbatively to first order in an expansion around the Markovian case \\beta = 1/2. Using the exact result \\theta_S = 1 - \\beta, accurate upper and lower bounds on \\theta_0 can be derived which show, in particular, that \\theta_0 \\geq (1 - \\beta)^2/\\beta for small \\beta.