Simulation of some spatial hard core models, and the complete packing problem

Abstract

A commonly used model for spatial point patterns exhibiting inhibition between points is the hard core model, in which the points of the pattern may be regarded as being the centres of non-overlapping discs of fixed diameter. It is often necessary in spatial statistics to be able to simulate many realisations of various models for point patterns. In this paper, the computer simulation of two stochastic hard core models is considered. The first of these, the Kelly-Ripley model, is simulated using a spatial birth and death process, while, for the second, the “SSI” model, a birth process is used. In each case, a new method of simulation, which is considerably faster than the existing method at high densities of discs, is described. Both algorithms use the Dirichlet tessellation of the points. The properties of the algorithm for simulating the SSI process make it possible to investigate random sequential packing of discs in a rectangular container. An estimate of the limiting packing density at complete packing of the rectangle is obtained; this should be an improvement on previous estimates.

Keywords:

• Spatial pattern
• hard core process
• birth and death process
• Dirichlet tessellation
• random sequential packing
• complete packing