TY - JOUR

T1 - Realizability of point processes

AU - Kuna, T.

AU - Lebowitz, J. L.

AU - Speer, E. R.

N1 - Funding Information:
Acknowledgements We thank E. Caglioti, R. Fernandez, G. Gallavotti, I. Kanter, Yu.G. Kondratiev, P.P. Mitra, J.K. Percus, F. Stillinger, S. Torquato, A. van Enter, and S.R.S. Varadhan for valuable comments. We also thank the IHES, and J.L.L. and E.R.S. thank the IAS for hospitality during the course of this work. The work of T.K. was supported by the A. v. Humboldt Foundation and SFB 701. The work of J.L.L. and T.K. was supported by NSF Grant DMR-0442066 and AFOSR Grant AF-FA9550-04. We also thank DIMACS and its supporting agencies. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

PY - 2007/11

Y1 - 2007/11

N2 - 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.

AB - 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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U2 - 10.1007/s10955-007-9393-y

DO - 10.1007/s10955-007-9393-y

M3 - Article

AN - SCOPUS:35348834133

VL - 129

SP - 417

EP - 439

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3

ER -