  We investigate Anderson localization of two particles moving in a
two-dimensional (2D) disordered lattice and coupled by contact interactions.
Based on transmission-amplitude calculations for relatively large strip-shaped
grids, we find that all pair states are localized in lattices of infinite size.
In particular, we show that previous claims of an interaction-induced mobility
edge are biased by severe finite-size effects. The localization length of a
pair with zero total energy exhibits a nonmonotonic behavior as a function of
the interaction strength, characterized by an exponential enhancement in the
weakly interacting regime. Our findings also suggest that the many-body
mobility edge of the 2D Anderson-Hubbard model disappears in the zero-density
limit, irrespective of the (bosonic or fermionic) quantum statistics of the
particles.
