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).