Abstract
Unlike in the classical framework of Gibbs point processes (usually acting on the complete graph), in the context of nearest-neighbour Gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property. This paper introduces domain-wise (but not pointwise) inhibition stationary Gibbs models based on some tailor-made Delaunay subgraphs. All of them are subgraphs of the R-local Delaunay graph, defined as the Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circles of radii greater than some large positive real value R. The usual relative compactness criterion for point processes needed for the existence result is directly derived from the Ruelle-bound of the correlation functions. Furthermore, assuming only the nonnegativeness of the energy function, we have managed to prove the existence of the existence of R-local Delaunay stationary Gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy.
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Baddeley, A.J., Møller, J.: Nearest-neighbour Markov point processes and random sets. Int. Stat. Rev. 57(2), 89–121 (1989)
Bertin, E., Billiot, J.-M., Drouilhet, R.: Existence of Delaunay pairwise Gibbs point processes with superstable component. J. Stat. Phys. 95, 719–744 (1999)
Bertin, E., Billiot, J.-M., Drouilhet, R.: Existence of “nearest-neighbour” Gibbs point models. Adv. Appl. Prob. 31, 895–909 (1999)
Bertin, E., Billiot, J.-M., Drouilhet, R.: Phase transition in nearest-neighbour continuum Potts models. J. Stat. Phys. 114(1/2), 79–100 (2004)
Connelly, R., Rybnikov, K., Volkov, S.: Percolation of the loss of tension in an infinite triangular lattice. J. Stat. Phys. 105(1/2), 143–171 (2001)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes. Springer Series in Statistics. Springer, New York (1988)
Georgii, H.-O.: Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31–51 (1976)
Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528 (1996)
Georgii, H.-O., Zagrebnov, V.A.: On the interplay of magnetic and molecular forces in Curie-Weiss ferrofluid models. J. Stat. Phys. 93, 79–107 (1998)
Georgii, H.-O., Zessin, H.: Large deviations and the maximum entropy principle for marked point random fields. Probab. Theor. Relat. Fields 96, 177–204 (1993)
Kendall, W.S., van Lieshout, M.N.M., Baddeley, A.J.: Quermass-interaction processes: conditions for stability. Adv. Appl. Probab. 31, 315–342 (1999)
Menshikov, M., Rybnikov, K., Volkov, S.: The loss of tension in an infinite membrane with holes distributed according to a Poisson law. Adv. Appl. Probab. 34(2), 292–312 (2002)
Preston, C.J.: Random Fields, vol. 534. Springer, Berlin (1976)
Ruelle, D.: Statistical Mechanics. Benjamin, New York (1969)
Ruelle, D.: Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)
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In memory of Etienne Bertin.
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Bertin, E., Billiot, JM. & Drouilhet, R. R-Local Delaunay Inhibition Model. J Stat Phys 132, 649–667 (2008). https://doi.org/10.1007/s10955-008-9565-4
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DOI: https://doi.org/10.1007/s10955-008-9565-4