T. Kuna, J. L. Lebowitz, E. R. Speer
Realizability of point processes
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ABSTRACT. There are various situations in which it is natural to ask
whether a given collection of $k$ functions,
$\rho_j(\r_1,\ldots,\r_j)$, $j=1,\ldots,k$, 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 $X=\RR^d$,
$X=\ZZ^d$, and $X$ an arbitrary finite set. In particular, we
extend a result by Ambartzumian and Sukiasian showing realizability
at sufficiently small densities $\rho_1(\r)$. 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 $\ZZ$;
there is a gap between our best upper bound on possible values of
$\rho$ and the largest $\rho$ for which realizability can be
established.