  We study classical hard-core dimer models on the square lattice with links
extending beyond nearest-neighbors. Numerically, using a directed-loop Monte
Carlo algorithm, we find that, in the presence of longer dimers preserving the
bipartite graph structure, algebraic correlations persist. While the
confinement exponent for monomers drifts, the leading decay of dimer
correlations remains 1/r^2, although the logarithmic peaks present in the dimer
structure factor of the nearest-neighbour model vanish. By contrast, an
arbitrarily small fraction of next-nearest-neighbor dimers leads to the onset
of exponential dimer correlations and deconfinement. We discuss these results
in the framework of effective theories, and provide an approximate but accurate
analytical expression for the dimer correlations.
