Isomorphisms for random sequential packing on lattices

Abstract

Recently Nakamura (1986) reported results for two classes of random sequential packing problems on n*n square lattices. Non-overlapping squares covering a2 cells are placed on the lattice either A allowing, or B forbidding contact. Here the author demonstrates an exact isomorphism between these two classes, as well as a broader range of models (and their d-dimensional analogues). Nakamura first took the n to infinity infinite-lattice limit and then analysed convergence of the packing fraction, p (i.e. saturation or jamming coverage), to the a to infinity continuum limit (where A and B coincide). The author obtains p values for model A with n= infinity from the generalised Palasti conjecture (using exact d=1 values), and develops a corresponding a to infinity asymptotic expansion; the corresponding p values for model B are obtained from isomorphism arguments. These facilitate analysis of Nakamura's results. Isomorphisms and continuum limit behaviour are also discussed for processes on other lattices.