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 $0<\mathrm{}\beta <1\mathrm{}$; for $\beta =\frac{1}{2}$ 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 ${\theta }_S=1\mathrm{}-\beta$. The exponent ${\theta }_0$ for the flat initial condition appears to be nontrivial. We prove that ${\theta }_0\to \infty$ for $\beta \to 0$, ${\theta }_0>~{\theta }_S$ for $\beta <$ $\frac{1}{2}$ and ${\theta }_0<~{\theta }_S$ for $\beta >$ $\frac{1}{2}$, and calculate ${\theta }_{0,S}$ perturbatively to first order in an expansion around the Markovian case $\beta =$ $\frac{1}{2}$. Using the exact result ${\theta }_S=1\mathrm{}-\beta$, accurate upper and lower bounds on ${\theta }_0$ can be derived which show, in particular, that ${\theta }_0>~(1\mathrm{}-\beta {)}^2/\beta$ for small $\beta$.