abstract: The big jump principle is a well established mathematical result for sums of independent and identically distributed random variables extracted from a fat tailed distribution. It states that the tail of the distribution of the sum is the same as the distribution of the largest summand. In practice, it means that when in a stochastic process the relevant quantity is a sum of variables, the mechanism leading to rare events is peculiar: instead of being caused by a set of many small deviations all in the same direction, one jump, the biggest of the lot, provides the main contribution to the rare large fluctuation. We extend here the big jump principle to a wide class of physical processes, which can be recasted in terms of transport and L\'evy walks. We analyze a L\'evy walk, a model diffusion for laser cold atoms, diffusion on a L\'evy-Lorentz gas and a correlated model of anomalous diffusion, showing that the principle holds and using it to predict rare fluctuations.
Joint work with Alessandro Vezzani, and Raffaella Burioni.