On the Construction of Particle Distributions with Specified Single and Pair Densities

TL;DR

This paper explores the conditions under which particle distributions with specified densities and pair correlations can exist, providing criteria for different types of point processes and illustrating with examples.

Contribution

It establishes necessary and sufficient conditions for the existence of certain particle distributions, including renewal and determinantal point processes, with explicit criteria and examples.

Findings

Necessary and sufficient conditions for renewal processes in 1D.

Maximum density for Poisson process dilution with given pair correlation.

Conditions for determinantal point processes in all dimensions.

Abstract

We discuss necessary conditions for the existence of probability distribution on particle configurations in $d$-dimensions i.e. a point process, compatible with a specified density $\rho$ and radial distribution function $g({\bf r})$. In $d=1$ we give necessary and sufficient criteria on $\rho g({\bf r})$ for the existence of such a point process of renewal (Markov) type. We prove that these conditions are satisfied for the case $g(r) = 0, r < D$ and $g(r) = 1, r > D$, if and only if $\rho D \leq e^{-1}$: the maximum density obtainable from diluting a Poisson process. We then describe briefly necessary and sufficient conditions, valid in every dimension, for $\rho g(r)$ to specify a determinantal point process for which all $n$-particle densities, $\rho_n({\bf r}_1, ..., {\bf r}_n)$, are given explicitly as determinants. We give several examples.