Big Jump Principle for First Passage Times

First passage times in disordered media, such as those involving charge carriers in amorphous semiconductors, contaminants in hydrogeological systems, or single macro-molecules in the the cell environment, are often described by time scale-free processes. We study the statistical properties of the first passage time of biased processes in different models, and employ the big jump principle that shows the dominance of the maximum trapping time on the first passage time. This leads to a scenario unlike that found for ordered systems, where many small short lived jumps lead to the completion of the first passage process. Inspired by the restart paradigm, we demonstrate that the removal of the maximum significantly expedites the first passage process. As the disorder increases, the system enters a phase where the removal shows a dramatic effect, the system is shown to exhibit a dynamical phase transition.