Asymptotic packing densities for two-dimensional lattice models

Blocks of nodes are randomly packed on torus shaped lattices, and estimates of some random packing densities are obtained by repeated simulation. Using nonlinear regression to fit the estimated densities, an asymptotic approximation formula is obtained for the packing densities which is similar to Mackenzie’s result in the analogous one-dimensional setting. Upon taking a limit, the formula yields an estimate of the analog of Renyi’s asymptotic packing constant which is in close agreement with some values that have been proposed by others.