Abstract: | Brownian motion is a Gaussian process described by the central limit theorem. However, exponential decays of the positional probability density function P(X,t) of packets of spreading random walkers, were observed in numerous situations that include glasses, live cells and bacteria suspensions. We will show that such exponential behavior is generally valid in a large class of problems of transport in random media. By extending the Large Deviations approach for a continuous time random walk we uncover a general universal behavior for the decay of the density. It is found that fluctuations in the number of steps of the random walker, performed at finite time, lead to an exponential decay (with logarithmic corrections) of P(X,t), known as Laplace tails. We will display how this Laplace universality is affected by the application of external force and how it is terminated when the rare events are determined not by the accumulation of different events but rather by a singular occurrence. |