  We propose a realization of the one-dimensional random dimer model and
certain N-leg generalizations using cold atoms in an optical lattice. We show
that these models exhibit multiple delocalization energies that depend strongly
on the symmetry properties of the corresponding Hamiltonian and we provide
analytical and numerical results for the localization length as a function of
energy. We demonstrate that the N-leg systems possess similarities with their
1D ancestors but are demonstrably distinct. The existence of critical
delocalization energies leads to dips in the momentum distribution which serve
as a clear signal of the localization-delocalization transition. These momentum
distributions are different for models with different group symmetries and are
identical for those with the same symmetry.
