The problem of finding the minimum-energy configurations of particles
on a square lattice, subject to short-ranged repulsive interactions, i
s studied analytically. The study is relevant to charge-ordered states
of interacting fermions, as described by the spinless Falicov-Kimball
model. A simple model is introduced, corresponding to the limit of a
two-body potential which is a very rapidly decreasing convex function
of distance. Its ground states are found rigorously for certain densit
y ranges, including half filling. These agree with known properties of
neutral ground states of the large-U Falicov-Kimball model, suggestin
g a characterization of the latter as ''most homogeneous'' configurati
ons. For lower densities, a family of ground stales is found having th
e novel property that they are aperiodic even when the particle densit
y is a rational number. In some cases, local phase separation suggests
an inherent sensitivity to the detailed form of the interaction poten
tial.