Persistence exponents for fluctuating interfaces

Abstract

Numerical and analytic results for the exponent $\ensuremath{\theta}$ 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 $\ensuremath{\beta}$, with $0<\ensuremath{\beta}<1$; for $\ensuremath{\beta}=\frac{1}{2}$ 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 ${\ensuremath{\theta}}_{S}=1\ensuremath{-}\ensuremath{\beta}$. The exponent ${\ensuremath{\theta}}_{0}$ for the flat initial condition appears to be nontrivial. We prove that ${\ensuremath{\theta}}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$ for $\ensuremath{\beta}\ensuremath{\rightarrow}0$, ${\ensuremath{\theta}}_{0}>~{\ensuremath{\theta}}_{S}$ for $\ensuremath{\beta}<$$\frac{1}{2}$ and ${\ensuremath{\theta}}_{0}<~{\ensuremath{\theta}}_{S}$ for $\ensuremath{\beta}>$$\frac{1}{2}$, and calculate ${\ensuremath{\theta}}_{0,S}$ perturbatively to first order in an expansion around the Markovian case $\ensuremath{\beta}=$$\frac{1}{2}$. Using the exact result ${\ensuremath{\theta}}_{S}=1\ensuremath{-}\ensuremath{\beta}$, accurate upper and lower bounds on ${\ensuremath{\theta}}_{0}$ can be derived which show, in particular, that ${\ensuremath{\theta}}_{0}>~(1\ensuremath{-}\ensuremath{\beta}{)}^{2}/\ensuremath{\beta}$ for small $\ensuremath{\beta}$.