Journal of Statistical Physics

, Volume 129, Issue 3, pp 417–439 | Cite as

Realizability of Point Processes

  • T. Kuna
  • J. L. Lebowitz
  • E. R. Speer


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 ρ j ’s for this to be true. Our primary examples are X=ℝ d , X=ℤ d , and X an arbitrary finite set. In particular, we extend a result by Ambartzumian and Sukiasian showing realizability at sufficiently small densities ρ 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 ρ and translation invariant ρ 2 are specified on ℤ; there is a gap between our best upper bound on possible values of ρ and the largest ρ for which realizability can be established.


Correlation Function Point Process Stat Phys Pair Distribution Function Cluster Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Department of MathematicsRutgers UniversityNew BrunswickUSA
  3. 3.Department of PhysicsRutgers UniversityNew BrunswickUSA

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