  The behaviour of random quantum walks is known to be diffusive. Here we study
discrete time quantum walks in weak stochastic gauge fields. In the case of
position and spin dependent gauge field, we observe a transition from ballistic
to diffusive motion, with the probability distribution becoming Gaussian.
However, in contradiction to common belief, weak stochastic electric gauge
fields reveal the persistence of Bloch oscillations despite decoherence which
we demonstrate on simulations and prove analytically. The proposed models
provide insights into the interplay between randomness and coherent dynamics of
quantum walks.
