Realizability of point processes

Abstract

There are various situations in which it is natural to ask whether a given collection of $k$ functions, $ρ_j(\r_1,...,\r_j)$, $j=1,...,k$, defined on a set $X$, are the first $k$ correlation functions of a point process on $X$. Here we describe some necessary and sufficient conditions on the ${\rho }_j$'s for this to be true. Our primary examples are $X={\mathbb{R}}^d$, $X=\matbb{Z}^d$, and $X$ an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ${\rho }_1(\mathbf{r})$. Typically if any realizing process exists there will be many (even an uncountable number); in this case we prove, when $X$ is a finite set, the existence of a realizing Gibbs measure with $k$ body potentials which maximizes the entropy among all realizing measures. We also investigate in detail a simple example in which a uniform density $\rho$ and translation invariant ${\rho }_2$ are specified on $\mathbb{Z}$; there is a gap between our best upper bound on possible values of $\rho$ and the largest $\rho$ for which realizability can be established.

31 pages, 1 figure, gzipped tar file; Changes in Appendix A : comment about a improved cluster expansion by Fernandez et al.; result about non realizability of the bump function

References in corpus (2)

• The analyticity region of the hard sphere gas. Improved bounds
• The Existence of Pair Potential Corresponding to Specified Density and Pair Correlation

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