  The position $x(t)$ of a particle diffusing in a one-dimensional uncorrelated
and time dependent random medium is simply Gaussian distributed in the typical
direction, i.e. along the ray $x=v_0 t$, where $v_0$ is the average drift.
However, it has been found that it exhibits at large time sample to sample
fluctuations characteristic of the KPZ universality class when observed in an
atypical direction, i.e. along the ray $x = v t$ with $v \neq v_0$. Here we
show, from exact solutions, that in the moderate deviation regime $x - v_0 t
\propto t^{3/4}$ these fluctuations are precisely described by the finite time
KPZ equation, which thus describes the crossover between the Gaussian typical
regime and the KPZ fixed point regime for the large deviations. This confirms
heuristic arguments given in [2]. These exact results include the discrete
model known as the Beta RWRE, and a continuum diffusion. They predict the
behavior of the maximum of a large number of independent walkers, which should
be easier to observe (e.g. in experiments) in this moderate deviations regime.
