Introduction to the Theory of Gibbs Point Processes

  • David DereudreEmail author
Part of the Lecture Notes in Mathematics book series (LNM, volume 2237)


The Gibbs point processes (GPP) constitute a large class of point processes with interaction between the points. The interaction can be attractive, repulsive, depending on geometrical features whereas the null interaction is associated with the so-called Poisson point process. In a first part of this mini-course, we present several aspects of finite volume GPP defined on a bounded window in \(\mathbb {R}^d\). In a second part, we introduce the more complicated formalism of infinite volume GPP defined on the full space \(\mathbb {R}^d\). Existence, uniqueness and non-uniqueness of GPP are non-trivial questions which we treat here with completely self-contained proofs. The DLR equations, the GNZ equations and the variational principle are presented as well. Finally we investigate the estimation of parameters. The main standard estimators (MLE, MPLE, Takacs-Fiksel and variational estimators) are presented and we prove their consistency. For sake of simplicity, during all the mini-course, we consider only the case of finite range interaction and the setting of marked points is not presented.



The author thanks P. Houdebert, A. Zass and the anonymous referees for the careful reading and the interesting comments. This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01), the CNRS GdR 3477 GeoSto and the ANR project PPP (ANR-16-CE40-0016).


  1. 1.
    A. Baddeley, D. Dereudre, Variational estimators for the parameters of Gibbs point process models. Bernoulli 19(3), 905–930 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A. Baddeley, R. Turner, Practical maximum pseudolikelihood for spatial point patterns (with discussion). Aust. N. Z. J. Stat. 42(3), 283–322 (2000)MathSciNetCrossRefGoogle Scholar
  3. 3.
    A.J. Baddeley, M.N.M. van Lieshout, Area-interaction point processes. Ann. Inst. Stat. Math. 47(4), 601–619 (1995)MathSciNetCrossRefGoogle Scholar
  4. 4.
    A. Baddeley, P. Gregori, J. Mateu, R. Stoica, D. Stoyan, Case Studies in Spatial Point Process Models. Lecture Notes in Statistics, vol. 185 (Springer, New York, 2005)Google Scholar
  5. 5.
    J. Besag, Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B 36, 192–236 (1974). With discussion by D. R. Cox, A. G. Hawkes, P. Clifford, P. Whittle, K. Ord, R. Mead, J. M. Hammersley, and M. S. Bartlett and with a reply by the authorGoogle Scholar
  6. 6.
    J.-M. Billiot, J.-F. Coeurjolly, R. Drouilhet, Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes. Electron. J. Stat. 2, 234–264 (2008)MathSciNetCrossRefGoogle Scholar
  7. 7.
    J.T. Chayes, L. Chayes, R. Kotecký, The analysis of the Widom-Rowlinson model by stochastic geometric methods. Commun. Math. Phys. 172(3), 551–569 (1995)MathSciNetCrossRefGoogle Scholar
  8. 8.
    S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, 3rd edn. (Wiley, Chichester, 2013)CrossRefGoogle Scholar
  9. 9.
    J.-F. Coeurjolly, D. Dereudre, R. Drouilhet, F. Lavancier, Takacs-Fiksel method for stationary marked Gibbs point processes. Scand. J. Stat. 39(3), 416–443 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes. Vol. I. Elementary Theory and Methods. Probability and Its Applications (New York), 2nd edn. (Springer, New York, 2003).Google Scholar
  11. 11.
    D. Dereudre, Diffusion infini-dimensionnelles et champs de Gibbs sur l’espace des trajectoires continues. PhD, Ecole polytechnique Palaiseau (2002)Google Scholar
  12. 12.
    D. Dereudre, The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains. Adv. Appl. Probab. 41(3), 664–681 (2009)MathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Dereudre, Variational principle for Gibbs point processes with finite range interaction. Electron. Commun. Probab. 21, Paper No. 10, 11 (2016)Google Scholar
  14. 14.
    D. Dereudre, P. Houdebert, Infinite volume continuum random cluster model. Electron. J. Probab. 20(125), 24 (2015)Google Scholar
  15. 15.
    D. Dereudre, F. Lavancier, Consistency of likelihood estimation for Gibbs point processes. Ann. Stat. 45(2), 744–770 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    D. Dereudre, R. Drouilhet, H.-O. Georgii, Existence of Gibbsian point processes with geometry-dependent interactions. Probab. Theory Relat. Fields 153(3–4), 643–670 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    D. Dereudre, F. Lavancier, K. Staňková Helisová, Estimation of the intensity parameter of the germ-grain quermass-interaction model when the number of germs is not observed. Scand. J. Stat. 41(3), 809–829 (2014)MathSciNetCrossRefGoogle Scholar
  18. 18.
    R.L. Dobrushin, E.A. Pecherski, A criterion of the uniqueness of Gibbsian fields in the noncompact case, in Probability Theory and Mathematical Statistics (Tbilisi, 1982). Lecture Notes in Mathematics, vol. 1021 (Springer, Berlin, 1983), pp. 97–110Google Scholar
  19. 19.
    T. Fiksel, Estimation of parametrized pair potentials of marked and nonmarked Gibbsian point processes. Elektron. Informationsverarb. Kybernet. 20(5–6), 270–278 (1984)MathSciNetzbMATHGoogle Scholar
  20. 20.
    H.-O. Georgii, Canonical Gibbs Measures. Lecture Notes in Mathematics, vol. 760 (Springer, Berlin, 1979). Some extensions of de Finetti’s representation theorem for interacting particle systemsGoogle Scholar
  21. 21.
    H.-O. Georgii, Large deviations and the equivalence of ensembles for Gibbsian particle systems with superstable interaction. Probab. Theory Relat. Fields 99(2), 171–195 (1994)MathSciNetCrossRefGoogle Scholar
  22. 22.
    H.-O. Georgii, Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, vol. 9, 2nd edn. (Walter de Gruyter, Berlin, 2011)Google Scholar
  23. 23.
    H.-O. Georgii, T. Küneth, Stochastic comparison of point random fields. J. Appl. Probab. 34(4), 868–881 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    H.-O. Georgii, H.J. Yoo, Conditional intensity and Gibbsianness of determinantal point processes. J. Stat. Phys. 118(1–2), 55–84 (2005)MathSciNetCrossRefGoogle Scholar
  25. 25.
    H.-O. Georgii, H. Zessin, Large deviations and the maximum entropy principle for marked point random fields. Probab. Theory Relat. Fields 96(2), 177–204 (1993)MathSciNetCrossRefGoogle Scholar
  26. 26.
    C.J. Geyer, J. Møller, Simulation procedures and likelihood inference for spatial point processes. Scand. J. Stat. 21(4), 359–373 (1994)MathSciNetzbMATHGoogle Scholar
  27. 27.
    X. Guyon, Random Fields on a Network. Probability and Its Applications (New York) (Springer, New York, 1995). Modeling, statistics, and applications, Translated from the 1992 French original by Carenne LudeñaGoogle Scholar
  28. 28.
    P. Hall, On continuum percolation. Ann. Probab. 13(4), 1250–1266 (1985)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J.L. Jensen, Asymptotic normality of estimates in spatial point processes. Scand. J. Stat. 20(2), 97–109 (1993)MathSciNetzbMATHGoogle Scholar
  30. 30.
    J.L. Jensen, H.R. Künsch, On asymptotic normality of pseudo likelihood estimates for pairwise interaction processes. Ann. Inst. Stat. Math. 46(3), 475–486 (1994)MathSciNetzbMATHGoogle Scholar
  31. 31.
    J.L. Jensen, J. Møller, Pseudolikelihood for exponential family models of spatial point processes. Ann. Appl. Probab. 1(3), 445–461 (1991)MathSciNetCrossRefGoogle Scholar
  32. 32.
    W.S. Kendall, J. Møller, Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Adv. Appl. Probab. 32(3), 844–865 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    O.K. Kozlov, Description of a point random field by means of the Gibbs potential. Uspehi Mat. Nauk 30(6(186)), 175–176 (1975)Google Scholar
  34. 34.
    J.L. Lebowitz, A. Mazel, E. Presutti, Liquid-vapor phase transitions for systems with finite-range interactions. J. Stat. Phys. 94(5–6), 955–1025 (1999)MathSciNetCrossRefGoogle Scholar
  35. 35.
    S. Mase, Uniform LAN condition of planar Gibbsian point processes and optimality of maximum likelihood estimators of soft-core potential functions. Probab. Theory Relat. Fields 92(1), 51–67 (1992)MathSciNetCrossRefGoogle Scholar
  36. 36.
    K. Matthes, J. Kerstan, J. Mecke, Infinitely Divisible Point Processes (Wiley, Chichester, 1978). Translated from the German by B. Simon, Wiley Series in Probability and Mathematical StatisticsGoogle Scholar
  37. 37.
    J. Mayer, E. Montroll, Molecular distributions. J. Chem. Phys. 9, 2–16 (1941)CrossRefGoogle Scholar
  38. 38.
    R. Meester, R. Roy, Continuum Percolation. Cambridge Tracts in Mathematics, vol. 119 (Cambridge University Press, Cambridge, 1996)Google Scholar
  39. 39.
    I.S. Molchanov, Consistent estimation of the parameters of Boolean models of random closed sets. Teor. Veroyatnost. i Primenen. 36(3), 580–587 (1991)MathSciNetzbMATHGoogle Scholar
  40. 40.
    J. Møller, Lectures on random Voronoı̆ tessellations. Lecture Notes in Statistics, vol. 87 (Springer, New York, 1994)Google Scholar
  41. 41.
    J. Møller, K. Helisová, Likelihood inference for unions of interacting discs. Scand. J. Stat. 37(3), 365–381 (2010)MathSciNetCrossRefGoogle Scholar
  42. 42.
    J. Møller, R.P. Waagepetersen, Statistical Inference and Simulation for Spatial Point Processes. Monographs on Statistics and Applied Probability, vol. 100 (Chapman & Hall/CRC, Boca Raton, 2004)Google Scholar
  43. 43.
    X. Nguyen, H. Zessin, Integral and differential characterizations Gibbs processes. Mathematische Nachrichten, 88(1), 105–115 (1979)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Y. Ogata, M. Tanemura, Likelihood analysis of spatial point patterns. J. R. Stat. Soc. Ser. B 46(3), 496–518 (1984)MathSciNetzbMATHGoogle Scholar
  45. 45.
    S. Poghosyan, D. Ueltschi, Abstract cluster expansion with applications to statistical mechanical systems. J. Math. Phys. 50(5), 053509, 17 (2009)MathSciNetCrossRefGoogle Scholar
  46. 46.
    C. Preston, Random fields. Lecture Notes in Mathematics, vol. 534 (Springer, Berlin, 1976)CrossRefGoogle Scholar
  47. 47.
    D. Ruelle, Statistical Mechanics: Rigorous Results (W. A. Benjamin, Inc., New York, 1969)zbMATHGoogle Scholar
  48. 48.
    D. Ruelle, Superstable interactions in classical statistical mechanics. Commun. Math. Phys. 18, 127–159 (1970)MathSciNetCrossRefGoogle Scholar
  49. 49.
    R. Takacs, Estimator for the pair-potential of a Gibbsian point process. Statistics, 17(3), 429–433 (1986)MathSciNetCrossRefGoogle Scholar
  50. 50.
    J. van den Berg, C. Maes, Disagreement percolation in the study of Markov fields. Ann. Probab. 22(2), 749–763 (1994)MathSciNetCrossRefGoogle Scholar
  51. 51.
    M.N.M. van Lieshout, Markov Point Processes and Their Applications (Imperial College Press, London, 2000)CrossRefGoogle Scholar
  52. 52.
    B. Widom, J.S. Rowlinson, New model for the study of liquid-vapor phase transitions. J. Chem. Phys. 52, 1670–1684 (1970)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.University LilleVilleneuve-d’AscqFrance

Personalised recommendations