POINT PROCESSES WITH A CLUSTERING TRANSITION

Abstract

summaryThe paper examines the properties of a simple pairwise interaction point process or distribution, when the potential is attractive. This leads to various degrees of clustering. Using measures such as the normalizing constant, number of close pairs and a generalized scan statistic, it is found that there are effectively three regimes for the model, determined by the relative magnitudes of numbers of points N, the point domain Vand strength of interaction v. As N v increases, abrupt transitions between these regimes occur with the middle “critical” one holding for only a relatively small range of N v‐values. It is possible to simulate the process reliably in one dimension, unlike the planar version, and confirmatory results are reported. A discrete version of the model, more susceptible to analysis, is introduced; it leads to the same conclusions and exhibits a criticality resembling, but distinct from, that in processes with phase transitions. Possible application of the discrete model to urbanization is described.