In this paper, we propose a new comparison tool for spatial homogeneity of
point processes, based on the joint examination of void probabilities and
factorial moment measures. We prove that determinantal and permanental
processes, as well as, more generally, negatively and positively associated
point processes are comparable in this sense to the Poisson point process of
the same mean measure. We provide some motivating results and preview further
ones, showing that the new tool is relevant in the study of macroscopic,
percolative properties of point processes. This new comparison is also implied
by the directionally convex (dcx ordering of point processes, which has
already been shown to be relevant to comparison of spatial homogeneity of point
processes. For this latter ordering, using a notion of lattice perturbation, we
provide a large monotone spectrum of comparable point processes, ranging from
periodic grids to Cox processes, and encompassing Poisson point process as
well. They are intended to serve as a platform for further theoretical and
numerical studies of clustering, as well as simple models of random point
patterns to be used in applications where neither complete regularity northe
total independence property are not realistic assumptions.Comment: 23 pages, 1 figure. This submission revisits and adds to ideas
concerning clustering and dcx ordering presented in arXiv:1105.4293.
Results on associated point process in Section 3.3 are new. arXiv admin note:
substantial text overlap with arXiv:1105.429
