Brownian motion is a Gaussian process described by the central limit theorem (CLT). However, exponential decay of the probability desnity function of packets of spreading random walkers, were observed in surprisingly many situations including tracers in glasses, live cells and bacteria suspensions. Extending large deviation theory we show that such exponential tails are generally valid, under mild conditions, and stem from fluctuations of the number of jumps performed by the particle. Unlike the CLT which is valid for long times and distances scaling only like the square root of t, here we consider large displacements yet finite measurement times, where the universal exponential tails are a rule. Then following experimental observations, we propose the Hitchhiker model to explain some aspects of motion of particles in the cell environment. Merging and breaking of macro-molecues  yields a fluctuating in the number of jumps made by the tracer particle, and hence the emergence of Laplace (=exponential) distributions. In this many body scenraio, the diffusion process is coupled to fragmantation and aggregation events, the tracking protocol applied to tracing particles is cruicial for the estimation of the dynamics.  A single particle has a statistical tendency to stick to large conglomeres hence its dynamics (for times that are long but not infinitely long) is slowed down if compared with an ensemble measurement performed on all the particles in the system. This simple effect has consequunces to the estimation of diffusion fields from single molecule tracking. 

[1] PRL (in press)  arXiv:1907.10002 [cond-mat.stat-mech] EB and S. Burov. 

[2] arXiv:1909.07189 [cond-mat.stat-mech] M. Hidalgo-Soria, and EB