Close to Close Packing

For various lattice gas models with nearest neighbor exclusion (and, in one case, second-nearest neighbor exclusion as well), we obtain lower bounds on m, the average number of particles on the nonexcluded lattice sites closest to a given particle. They are all of the form m/m_cp1–const·(N_cp/N–1), where N is the number of occupied sites, m_cp is the value of m at close packing, and N_cp is the value of N at close packing. An analogous result exists for hard disks in the plane.