Realizability of point processes

TL;DR

This paper establishes necessary and sufficient conditions for a collection of functions to be the correlation functions of a point process, extending existing results and analyzing realizability, especially for finite sets and specific examples.

Contribution

It provides a comprehensive set of criteria for realizability of correlation functions, extending prior work to new settings and including entropy-maximizing measures for finite sets.

Findings

Conditions for realizability on $ eal^d$, $z^d$, and finite sets.

Extension of Ambartzumian and Sukiasian's results to small densities.

Existence of entropy-maximizing Gibbs measures for finite sets.

Abstract

There are various situations in which it is natural to ask whether a given collection of $k$ functions, $\rho_j(\r_1,...,\r_j)$, $j=1,...,k$, 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.