Persistence exponents for fluctuating interfaces

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 Aat 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(O) for the flat initial condition appears to be nontrivial. We prove that theta(O)-->infinity for beta-->0, theta(O) greater than or equal to theta(S) for beta<1/2 and theta(O) less than or equal to theta(S) for beta>1/2, and calculate theta(O,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(O) can be derived which show, in particular. that theta(O) greater than or equal to(1-beta)(2)/beta for small beta.