Existence of 'nearest-neighbour' spatial Gibbs models

Advances in Applied Probability

Volumn 31, Issue 4, 1999, Pages 895-909

  Bertin, Etienne ^a   Billiot, Jean Michel ^a   Drouilhet, Rémy ^a  

^a UNIV GRENOBLE ALPES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

ASYMPTOTIC STABILITY; BOUNDARY CONDITIONS; COMPUTATIONAL GEOMETRY; FUNCTIONS; GIBBS FREE ENERGY; MATHEMATICAL MODELS; POISSON DISTRIBUTION; SET THEORY; THEOREM PROVING;

DELAUNAY TRIANGULATION; DOBRUSHIN-LANDFORD-RUELLE EQUATIONS; GIBBS POINT PROCESSES; LOCAL SPECIFICATIONS; MARKOV COMPONENT TYPE MODELS; NEAREST NEIGHBOUR INTERACTION; ORD MODEL; PRESTON THEOREM; STOCHASTIC GEOMETRY;

MARKOV PROCESSES;

EID: 85037958424     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0001867800009514     Document Type: Article

References (39)

1. BADDELEY, A. AND MØLLER, J. (1989). Nearest-neighbour Markov point processes and random sets. Int. Statist. Rev. 57, 89-121.
2. BERTIN, E., BILLIOT, J.-M. AND DROUILHET, R. (1999). Existence of Delaunay pairwise Gibbs point processes with superstable component. J. Statist. Phys. 95, 719-744.
3. BERTIN, E., BILLIOT, J.-M. AND DROUILHET, R. (1999). k-Nearest-neighbour Gibbs point processes. Markov Proc. Rel. Fields 5, 219-234.
4. BERTIN, E., BILLIOT, J.-M. AND DROUILHET, R. ( 1999). Spatial Delaunay Gibbs point processes. Stochastic Models 15, 181-199.
5. BOISSONNAT, J.-D. AND YVINEC, M. (1995). Géométrie Algorithmique. Ediscience International, Paris.
6. BOOTS, B. (1986). Using angular properties of Delaunay triangles to evaluate point patterns. Geog. Anal. 18, 250-260.
7. CHAYES. J., CHAYES, L. AND KOTECKY, R. (1995). The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172, 551-569.
8. DALEY, D. AND VERE-JONES, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.
9. DOBRUSHIN, R. (1969). Gibbsian random field. The general case. Func. Anal. Appl. 3, 22-28.
10. EDELSBRUNNER, H. (1988). Algorithms in Combinatorial Geometry. Springer, New York.
11. FEYNMAN, R. (1972). Statistical Mechanics. A Set of Lectures. W. A. Benjamin, Reading, MA.
12. GEORGII, H.-O. (1976). Canonical and grand canonical Gibbs states for continuum systems. Commun. Math. Phys. 48, 31-51.
13. GEORGII, H.-O. (1994). Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Prob. Theor. Rel. Fields 99, 171-195.
14. GEORGII, H.-O. AND HÄGGSTRÖM, O. (1996). Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507-528.
15. GLÖTZL, E. (1980). Locale Energien und Potentiale für Punkprozesse. Math. Nachr. 96, 195-206.
16. GRIMMET, G. (1995). The stochastic random-cluster process, and uniqueness of random cluster measures. Ann. Prob. 23, 1461-1510.
17. GRUBER, C. AND LEBOWITZ, J. (1975). Equilibrium states for classical systems. Commun. Math. Phys. 41, 11-18.
18. HÄGGSTRÖM, O., VAN LIESHOUT, M. AND MØLLER, J. (1996). Characterisation results and Markov chain Monte Carlo algorithms including exact simulation for spatial point processes. Technical Report R. 96-2040, Department of Mathematics, Aalborg University. To appear in Bernoulli.
19. VAN HOVE, L. (1949). Quelques propriétés générales de l'intégrale de configuration d'un système de particules avec interaction. Physica 15, 951-961.
20. KALLENBERG, O. (1983). Random Measures. Akademie-Verlag and Academic Press, Berlin, London.
21. KENDALL, W. (1990). A spatial Markov property for nearest-neighbour Markov point processes. J. Appl. Prob. 28, 767-778.
22. KENDALL, W. (1998). Perfect simulation for the area-interaction point process. In Probability Towards the Year 2000, eds L. Accardy and C. C. Heyde. Springer, New York, pp. 218-334.
23. KLEIN, D. (1982). Dobrushin uniqueness techniques and the decay of correlations in continuum statistical mechanics. Commun. Math. Phys. 86, 227-246.
24. KLEIN, D. (1984). Convergence of grand canonical Gibbs measures. Commun. Math. Phys. 92, 295-308.
25. LANFORD, O. AND RUELLE, D. (1969). Observables at infinity and states with support range correlations in statistical mechanics. Commun. Math. Phys. 13, 194-215.
26. LEBOWITZ, J. AND LIEB, E. (1972). Phase transition in continuum classical system with finite interactions. Phys. Lett. 39, 98-100.
27. MILES, R. (1970). On the homogeneous planar Poisson point process. Math. Biosci. 6, 85-127.
28. MØLLER, J. (1994). Discussion contribution. Scand. J. Statist. 21, 346-349.
29. MØLLER, J. (1994). Lectures on Random Voronoi Tessellations (Lecture Notes in Statist. 87). Springer, New York.
30. MØLLER, J. AND WAAGEPETERSEN, R. (1998). Markov connected component fields. Adv. Appl. Prob. 30, 1-35.
31. NGUYEN, X. AND ZESSIN, H. (1976). Integral and differential characterizations of the Gibbs process. Math. Nachr. 88, 105-115.
32. PREPARATA, F. AND SHAMOS, M. (1988). Computational Geometry, an Introduction. Springer, New York.
33. PRESTON, C. (1976). Random Fields. Springer, Berlin.
34. RIPLEY, B. (1977). Modelling spatial patterns (with discussion). J. R. Statist. Soc. B 39, 172-212.
35. RUELLE, D. (1969). Statistical Mechanics. W. R. Benjamin, New York.
36. RUELLE, D. (1970). Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127-159.
37. RUELLE, D. (1971). Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040-1041.
38. SIBSON, R. (1978). Locally equiangular triangulations. Computer J. 21, 243-245.
39. STOYAN, D., KENDALL, W. AND MECKE, J. (1995). Stochastic Geometry and its Applications, 2nd edn. John Wiley, Chichester, UK.