Résumé
Glass-forming liquids have only a modest tendency to crystallize and hence their dynamics can be studied even below the melting temperature. The relaxation dynamics of most of these liquids shows at a temperature$T_c$ , somewhat above the glass-transition temperature$T_g$ , a crossover, which indicates the conjunction of two different dynamical regimes. For temperatures slightly above$T_c$ , experiments and computer simulations have extensively probed this dynamics on the particle level and identified several universal scaling laws that are often compatible with theoretical predictions. Using large scale computer simulations we extend these studies to temperatures below$T_c$and find that the relaxation mechanism is qualitatively different from the one found at higher temperatures. We identify new scaling laws that allow to give a simple description of the relaxation dynamics at very low$T$ s. Specifically we reveal that the cage-escape process is related to rare but large particle displacements that give rise to a distinctive sub-diffusive power-law in the time correlation functions. This insight helps to advance our understanding on the relaxation dynamics of glass-forming systems at temperatures that are close to the experimental glass transition.