POISSON LIMITS FOR PAIRWISE AND AREA INTERACTION POINT PROCESSES

By S. Rao Jammalamadaka and Mathew D. Penrose.

Suppose n particles xi in a region of the plane (possibly representing biological individuals - trees, organisms) which is weighted, relative to an i.i.d. distribution in the plane, by a factor of exp (-S), where S denotes the sum over all distinct pairs i,j of the quanitity f (n(xi-xj)), with f a specified function of compact support. We obtain a Poisson process limit for the collection of rescaled interparticle distances as n becomes large. We also give corresponding results for the case of several types of particles, representing different species, and also for the area-interaction (Widom-Rowlinson) point process of interpenetrating spheres.

Advances in Applied Probability 32, 75-85 (2000).