  We examine the structure of the distribution of single particle displacements
(van-Hove function) in a broad class of materials close to glass and jamming
transitions. In a wide time window comprising structural relaxation, van-Hove
functions reflect the coexistence of slow and fast particles (dynamic
heterogeneity). The tails of the distributions exhibit exponential, rather than
Gaussian, decay. We argue that this behavior is universal in glassy materials
and should be considered the analog, in space, of the stretched exponential
decay of time correlation functions. We introduce a dynamical model that
describes quantitatively numerical and experimental data in supercooled
liquids, colloidal hard spheres and granular materials. The tails of the
distributions directly explain the decoupling between translational diffusion
and structural relaxation observed in glassy materials.
