Title: Fast rare events in exit times distributions of jump processes

Abstract: Rare events in the first-passage distributions of jump processes are capable of triggering anomalous reactions or series of events. Estimating their probability is particularly important when the jump probabilities have broad-tailed distributions, and rare events are therefore not so rare. We study three jump processes that are used to model a wide class of stochastic processes ranging from biology to transport in disordered systems, ecology and finance. We consider discrete time random-walks, continuous time random-walks and the L\'evy-Lorentz gas and determine the exact form of the scaling function for the probability distribution of fast rare events, in which the jump process exits in a very short time at a large distance opposite to the starting point. For this estimation we use the so called big jump principle, and we show that in the regime of fast rare events the exit time distributions are not exponentially suppressed, even in the case of normal diffusion. This implies that fast rare events are actually much less rare than predicted by the usual estimates of large deviations and can occur on timescales orders of magnitude shorter than expected. Our results are confirmed by extensive numerical simulations.