  Persistence probabilities of the interface height in (1+1)- and
(2+1)-dimensional atomistic, solid-on-solid, stochastic models of surface
growth are studied using kinetic Monte Carlo simulations, with emphasis on
models that belong to the molecular beam epitaxy (MBE) universality class. Both
the initial transient and the long-time steady-state regimes are investigated.
We show that for growth models in the MBE universality class, the nonlinearity
of the underlying dynamical equation is clearly reflected in the difference
between the measured values of the positive and negative persistence exponents
in both transient and steady-state regimes. For the MBE universality class, the
positive and negative persistence exponents in the steady-state are found to be
$\theta^S_{+} = 0.66 \pm 0.02$ and $\theta^S_{-} = 0.78 \pm 0.02$,
respectively, in (1+1) dimensions, and $\theta^S_{+} = 0.76 \pm 0.02$ and
$\theta^S_{-} =0.85 \pm 0.02$, respectively, in (2+1) dimensions. The noise
reduction technique is applied on some of the (1+1)-dimensional models in order
to obtain accurate values of the persistence exponents. We show analytically
that a relation between the steady-state persistence exponent and the dynamic
growth exponent, found earlier to be valid for linear models, should be
satisfied by the smaller of the two steady-state persistence exponents in the
nonlinear models. Our numerical results for the persistence exponents are
consistent with this prediction. We also find that the steady-state persistence
exponents can be obtained from simulations over times that are much shorter
than that required for the interface to reach the steady state. The dependence
of the persistence probability on the system size and the sampling time is
shown to be described by a simple scaling form.
