  In a growing number of strongly disordered and dense systems, the dynamics of
a particle pulled by an external force field exhibits super-diffusion. In the
context of glass forming systems, super cooled glasses and contamination
spreading in porous medium it was suggested to model this behavior with a
biased continuous time random walk. Here we analyze the plume of particles far
lagging behind the mean, with the single big jump principle. Revealing the
mechanism of the anomaly, we show how a single trapping time, the largest one,
is responsible for the rare fluctuations in the system. These non typical
fluctuations still control the behavior of the mean square displacement, which
is the most basic quantifier of the dynamics in many experimental setups. We
show how the initial conditions, describing either stationary state or
non-equilibrium case, persist for ever in the sense that the rare fluctuations
are sensitive to the initial preparation. To describe the fluctuations of the
largest trapping time, we modify Fr\'{e}chet's law from extreme value
statistics, taking into consideration the fact that the large fluctuations are
very different from those observed for independent and identically distributed
random variables.
