  We discuss the combined effects of overdamped motion in a quenched random
potential and diffusion, in one dimension, in the limit where the diffusion
coefficient is small. Our analysis considers the statistics of the mean
first-passage time $T(x)$ to reach position $x$, arising from different
realisations of the random potential: specifically, we contrast the median
$\bar T(x)$, which is an informative description of the typical course of the
dispersion, with the expectation value $\langle T(x)\rangle$, which is
dominated by rare events where there is an exceptionally high barrier to
diffusion. We show that at relatively short times the median $\bar T(x)$ is
explained by a 'flooding' model, where $T(x)$ is predominantly determined by
the highest barriers which is encountered before reaching position $x$. These
highest barriers are quantified using methods of extreme value statistics.
