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Existence of Gibbs point processes with stable infinite range interaction

Part of: Geometric probability and stochastic geometry Equilibrium statistical mechanics Special processes Stochastic processes

Published online by Cambridge University Press:  04 September 2020

David Dereudre  and
Thibaut Vasseur
David Dereudre*
Affiliation:
Université de Lille
Thibaut Vasseur*
Affiliation:
Université de Lille
*
*Postal address: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France. Email address: david.dereudre@univ-lille.fr
*Postal address: Université de Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France. Email address: david.dereudre@univ-lille.fr
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Abstract

We provide a new proof of the existence of Gibbs point processes with infinite range interactions, based on the compactness of entropy levels. Our main existence theorem holds under two assumptions. The first one is the standard stability assumption, which means that the energy of any finite configuration is superlinear with respect to the number of points. The second assumption is the so-called intensity regularity, which controls the long range of the interaction via the intensity of the process. This assumption is new and introduced here since it is well adapted to the entropy approach. As a corollary of our main result we improve the existence results by Ruelle (1970) for pairwise interactions by relaxing the superstabilty assumption. Note that our setting is not reduced to pairwise interaction and can contain infinite-range multi-body counterparts.

Keywords

DLR equationsentropy boundssuperstable interaction

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry 82B21: Continuum models (systems of particles, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 3 , September 2020 , pp. 775 - 791
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Copyright
© Applied Probability Trust 2020

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