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We examine the structure of the distribution of single particle displacements(van-Hove function) in a broad class of materials close to glass and jammingtransitions. In a wide time window comprising structural relaxation, van-Hovefunctions reflect the coexistence of slow and fast particles (dynamicheterogeneity). The tails of the distributions exhibit exponential, rather thanGaussian, decay. We argue that this behavior is universal in glassy materialsand should be considered the analog, in space, of the stretched exponentialdecay of time correlation functions. We introduce a dynamical model thatdescribes quantitatively numerical and experimental data in supercooledliquids, colloidal hard spheres and granular materials. The tails of thedistributions directly explain the decoupling between translational diffusionand structural relaxation observed in glassy materials.

Universal nature of particle displacements close to glass and jamming transitions