  Recently it has been shown that when an equation that allows so-called pulled
fronts in the mean-field limit is modelled with a stochastic model with a
finite number $N$ of particles per correlation volume, the convergence to the
speed $v^*$ for $N \to \infty$ is extremely slow -- going only as $\ln^{-2}N$.
However, this convergence is seen only for very high values of $N$, while there
can be significant deviations from it when $N$ is not too large. Pulled fronts
are fronts that propagate into an unstable state, and the asymptotic front
speed is equal to the linear spreading speed $v^*$ of infinitesimal
perturbations around the unstable state. In this paper, we consider front
propagation in a simple stochastic lattice model. The microscopic picture of
the front dynamics shows that for the description of the far tip of the front,
one has to abandon the idea of a uniformly translating front solution. The
lattice and finite particle effects lead to a ``halt-and-go'' type dynamics at
the far tip of the front, while the average front behind it ``crosses over'' to
a uniformly translating solution. In this formulation, the effect of
stochasticity on the asymptotic front speed is coded in the probability
distribution of the times required for the advancement of the ``foremost
occupied lattice site''. These probability distributions are obtained by
matching the solution of the far tip with the uniformly translating solution
behind in a mean-field type approximation, and the results for the probability
distributions compare well to the results of stochastic numerical simulations.
This approach allows one to deal with much smaller values of $N$ than it is
required to have the $\ln^{-2}N$ asymptotics to be valid.
