  Although time-dependent random media with short range correlations lead to
(possibly biased) normal tracer diffusion, anomalous fluctuations occur away
from the most probable direction. This was pointed out recently in 1D lattice
random walks, where statistics related to the 1D Kardar- Parisi-Zhang (KPZ)
universality class, i.e. the GUE Tracy Widom distribution, were shown to arise.
Here we provide a simple picture for this correspondence, directly in the
continuum as well as for lattice models, which allows to study arbitrary space
dimension and to predict a variety of universal distributions. In $d = 1$ we
predict and verify numerically the emergence of the GOE Tracy-Widom
distribution for the fluctuations of the transition probability. In $d = 3$ we
predict a phase transition from Gaussian fluctuations to 3D-KPZ type
fluctuations as the bias is increased. We predict KPZ universal distributions
for the arrival time of a first particle from a cloud diffusing in such media.
