  In biological, glassy, and active systems, various tracers exhibit
Laplace-like, i.e., exponential, spreading of the diffusing packet of
particles. The limitations of the central limit theorem in fully capturing the
behaviors of such diffusive processes, especially in the tails, have been
studied using the continuous time random walk model. For cases when the jump
length distribution is super-exponential, e.g., a Gaussian, we use large
deviations theory and relate it to the appearance of exponential tails. When
the jump length distribution is sub-exponential the packet of spreading
particles is described by the big jump principle. We demonstrate the
applicability of our approach for finite time, indicating that rare events and
the asymptotics of the large deviations rate function can be sampled for large
length scales within a reasonably short measurement time.
