  This paper investigates quantum diffusion of matter waves in two-dimensional
random potentials, focussing on expanding Bose-Einstein condensates in
spatially correlated optical speckle potentials. Special care is taken to
describe the effect of dephasing, finite system size, and an initial momentum
distribution. We derive general expressions for the interference-renormalized
diffusion constant, the disorder-averaged probability density distribution, the
variance of the expanding atomic cloud, and the localized fraction of atoms.
These quantities are studied in detail for the special case of an
inverted-parabola momentum distribution as obtained from an expanding
condensate in the Thomas-Fermi regime. Lastly, we derive quantitative criteria
for the unambiguous observation of localization effects in a possible 2D
experiment.
