Propagation in quantum walks and relativistic diffusions

Abstract: Propagation in quantum walks is revisited by showing that very general 1D discrete-time quantum walks with time- and space-dependent coefficients can be described, at the continuous limit, by Dirac fermions coupled to electromagnetic fields. Short-time propagation is also established for relativistic diffusions by presenting new numerical simulations of the Relativistic Ornstein-Uhlenbeck Process. A geometrical generalization of Fick's law is also obtained for this process. The results suggest that relativistic diffusions may be realistic models of decohering or random quantum walks. Links with general relativity and geometrical flows are also mentioned.