Realizability of point processes

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

Research output: Contribution to journalArticlepeer-review

36 Scopus citations


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=ℝd , X=ℤd , 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.

Original languageEnglish (US)
Pages (from-to)417-439
Number of pages23
JournalJournal of Statistical Physics
Issue number3
StatePublished - Nov 2007

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics


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