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Particle hopping is a common feature in heterogeneous media. We explore suchmotion by using the widely applicable formalism of the continuous time randomwalk and focus on the statistics of rare events. Numerous experiments haveshown 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 ofexponential distribution of particle hops is crossed. While the mean hop is notdiverging and can attain a finite value; the transition itself is critical. Theexponential universality of rare events arises due to the contribution of allthe different states occupied during the process. Once the reported thresholdis crossed, a single large event determines the statistics. In this realm, thebig jump principle replaces the large deviation principle, and the spatial partof the decay is unaffected by the temporal properties of rare events.

Universal to Non-Universal Transition of the statistics of Rare Events During the Spread of Random Walks