Journal of Statistical Physics, Vol. 63, Nos. 5/6, 1991. p. 1163-1176.
We describe a search for solidlike singlet distribution functions in a system of hard spheres. The procedure, which is based on Widom's relation between the activity and the density in a nonuniform fluid, is applied to a sequence of hardcore lattice gases with increasingly extended interactions. When the system is defined on a Bethe lattice we obtain exact solutions for arbitrary external field and size of the hard core. This includes the limit in which the number of excluded neighbors goes to infinity while the lattice spacing is made to vanish. The study of the first few members in this family of models suggests the existence of an infinite sequence, beginning with the next-nearest-neighbor problem, of first-order sublattice ordering transitions occurring before close packing and at zero field. The periodic solutions for the density originate at bifurcation points located at uniform close packing.