06-369 T. Kuna, J. L. Lebowitz, E. R. Speer
Realizability of point processes (397K, pdf) Dec 23, 06
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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.

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