  Particle hopping is a common feature in heterogeneous media. We explore such
motion by using the widely applicable formalism of the continuous time random
walk and focus on the statistics of rare events. Numerous experiments have
shown that the decay of the positional probability density function P (X, t),
describing the statistics of rare events, exhibits universal exponential decay.
We show that such universality ceases to exist once the threshold of
exponential distribution of particle hops is crossed. While the mean hop is not
diverging and can attain a finite value; the transition itself is critical. The
exponential universality of rare events arises due to the contribution of all
the different states occupied during the process. Once the reported threshold
is crossed, a single large event determines the statistics. In this realm, the
big jump principle replaces the large deviation principle, and the spatial part
of the decay is unaffected by the temporal properties of rare events.
