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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, 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.

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501, Bielefeld, Germany

    T. Kuna

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. L. Lebowitz & E. R. Speer

  3. Department of Physics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. L. Lebowitz

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  1. T. Kuna
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  2. J. L. Lebowitz
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  3. E. R. Speer
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Correspondence to T. Kuna.

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Kuna, T., Lebowitz, J.L. & Speer, E.R. Realizability of Point Processes. J Stat Phys 129, 417–439 (2007). https://doi.org/10.1007/s10955-007-9393-y

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