Brownian motion is a Gaussian process described by the central limit theorem. How- ever, 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 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 exponential decay (with logarithmic corrections) of $P(X, t)$. This universal behavior also holds for short times, a fact that makes experimental observations readily achievable. Time permitting we then formulate the hitchhiker model where interacting molecules form aggregates, that lead to fluctuations in the diffusion field, and a many body mechanism for the exponential tails .
 E. Barkai and Stas Burov, Packets of diffusing particles exhibit universal exponential tails, Phys. Rev. Lett. 124, 060603 (2020).
 M. Hidalgo-Soria, and E. Barkai, The Hitchhiker model for Laplace diffusion processes in the cell environment, Phys. Rev. E 102, 012109 (2020).