Probability Towards 2000

The area of random dynamical systems (henceforth abbreviated as ‘RDS’) can be superficially described as the ‘intersection’ of stochastic processes with dynamical systems. It is an example for the fact that a symbiosis of two mathematical disciplines at the right moment amounts to opening a scientific gold mine, both conceptually and as far as significant applications are concerned.

Because so many random processes arising in stochastic geometry are quite intractable to analysis, simulation is an important part of the stochastic geometry toolkit. Typically, a Markov point process such as the area-interaction point process is simulated (approximately) as the long-run equilibrium distribution of a (usually reversible) Markov chain such as a spatial birth-and-death process. This is a useful method, but it can be very hard to be precise about the length of simulation required to ensure that the long-run approximation is good. The splendid idea of Propp and Wilson [17] suggests a way forward: they propose a coupling method which delivers exact simulation of equilibrium distributions of (finite-state-space) Markov chains. In this paper their idea is extended to deal with perfect simulation of attractive area-interaction point processes in bounded windows. A simple modification of the basic algorithm is described which provides perfect simulation of the repulsive case as well (which being nonmonotonic might have been thought out of reach). Results from simulations using a C computer program are reported; these confirm the practicality of this approach in both attractive and repulsive cases. The paper concludes by mentioning other point processes which can be simulated perfectly in this way, and by speculating on useful future directions of research. Clearly workers in stochastic geometry should now seek wherever possible to incorporate the Propp and Wilson idea in their simulation algorithms.