  We report the results of zero temperature quantum Monte Carlo simulations and
zero temperature mean-field calculations of the attractive Hubbard model on
chains, ladders, and square lattices. We investigated the predictability of the
BCS approximation, the dimensional cross-over of the pairing correlation
function from one to two dimensions as a function of the ladder width, and the
scaling of these correlations to the thermodynamic limit of the two-dimensional
model. We found that the BCS wave function is quantitatively correct only for
small values of $U/t$. For the system sizes, electron fillings, and interaction
strengths studied, we never saw the dimensional cross-over. In general our
ability to achieve the dimensional cross-over and accurate scaling to the
thermodynamic limit was limited by the size of the systems we could simulate.
For these sizes, although we saw the necessary signature of ODLRO, the
properties of the model did not vary monotonically with increasing system size
because of shell effects. We contrast this situation with the dimensional
cross-over and scaling to the thermodynamic limit of the Ising model.
