Random walk on the randomly-oriented Manhattan lattice

In the randomly-oriented Manhattan lattice, every line in Zd is assigned a uniform random direction. We consider the directed graph whose vertex set is Zd and whose edges connect nearest neighbours, but only in the direction fixed by the line orientations. Random walk on this directed graph chooses uniformly from the d legal neighbours at each step. We prove that this walk is superdiffusive in two and three dimensions. The model is diffusive in four and more dimensions.