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Large deviations and the maximum entropy principle for marked point random fields
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  • Published: June 1993

Large deviations and the maximum entropy principle for marked point random fields

  • Hans-Otto Georgii1 &
  • Hans Zessin2 

Probability Theory and Related Fields volume 96, pages 177–204 (1993)Cite this article

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Summary

We establish large deviation principles for the stationary and the individual empirical fields of Poisson, and certain interacting, random fields of marked point particles in ℝd. The underlying topologies are induced by a class of not necessarily bounded local functions, and thus finer than the usual weak topologies. Our methods yield further that the limiting behaviour of conditional Poisson distributions, as well as certain distributions of Gibbsian type, is governed by the maximum entropy principle. We also discuss various applications and examples.

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References

  1. Csiszár, I.: Sanov property, generalizedI-projection and a conditional limit theorem. Ann. Probab.12, 768–793 (1984)

    Google Scholar 

  2. Daley, D.J., Vere-Jones, D.: An introduction to the theory of point processes. Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  3. Deuschel, J.D., Stroock, D.W., Zessin, H.: Microcanonical distributions for lattice gases. Commun. Math. Phys.139, 83–101 (1991)

    Google Scholar 

  4. Ellis, R.S., Newman, C.M.: Necessary and sufficient conditions for the GHS inequality with applications to probability and analysis. Trans. Am. Math. Soc.237, 83–99 (1978)

    Google Scholar 

  5. Föllmer, H.: Random fields and diffusion processes. In: Hennequin, P.L. (ed.) Ecole d'Eté de Probabilités de Saint-Flour XV-XVII. (Lect. Notes Math., vol. 1362, pp. 101–203) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  6. Fritz, J.: Generalization of McMillan's theorem to random set functions. Stud. Sci. Math. Hung.5, 369–394 (1970)

    Google Scholar 

  7. Georgii, H.O.: Gibbs measures and phase transitions. Berlin: de Gruyter 1988

    Google Scholar 

  8. Georgii, H.O.: Large deviations and maximum entropy principle for interacting random fields onZ d. Ann. Probab. (to appear, 1993)

  9. Georgii, H.O.: Large deviations for hard-core particle systems. In: Kotecky, R. (ed.) Proceedings of the 1992 Prague Workshop on Phase Transitions. Singapore: World Scientific 1993

    Google Scholar 

  10. Krickeberg, K.: Processes ponctuels en statistique. In: Hennequin, P.L. (ed.) Ecole d'Eté de Probabilités de Saint-Flour X. (Lect. Notes Math., vol. 929, pp. 206–313) Berlin Heidelberg New York: Springer 1980

    Google Scholar 

  11. Matthes, K., Kerstan, J., Mecke, J.: Infinitely divisible point processes. Chichester: Wiley 1978

    Google Scholar 

  12. Mecke, J.: Stationäre zufällige Maße auf lokalkompakten abelschen Gruppen. Z. Wahrscheinlichkeitstheor. Verw. Geb.9, 36–58 (1967)

    Google Scholar 

  13. Nguyen, X.X., Zessin, H.: Ergodic theorems for spatial processes. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 133–158 (1979)

    Google Scholar 

  14. Olla, S., Varadhan, S.R.S., Yau, H.T.: Hydrodynamic limit for a Hamiltonian system with a weak noise (Preprint 1992)

  15. Papangelou, F.: On the entropy rate of stationary point processes and its discrete approximation. Z. Wahrscheinlichkeitstheor. Verw. Geb.44, 191–211 (1978)

    Google Scholar 

  16. Rockafellar, R.T.: Convex analysis, Princeton: Princeton University Press 1970

    Google Scholar 

  17. Roelly, S., Zessin, H.: The equivalence of equilibrium principles in statistical mechanics and some applications to large particle systems. Expos. Math. (to appear, 1993)

  18. Ruelle, D.: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett.27, 1040–1041 (1971)

    Google Scholar 

  19. Varadhan, S.R.S.: Large deviations and applications. In: Hennequin, P.L. (ed.) Ecole d'Eté de Probabilités de Saint Flour XV–XVII. (Lect. Notes Math., vol. 1362, pp. 1–49) Berlin Heidelberg New York: Springer 1988

    Google Scholar 

  20. Widom, B., Rowlinson, J.S.: New model for the study of liquid-vapor phase trasitions. J. Chem. Phys.52, 1670–1684 (1970)

    Google Scholar 

  21. Zessin, H.: Boltzmann's principle for Brownian motion on the torus. In: Blanchard, Ph. (ed.) Dynamics of complex and irregular systems. Singapore: World Scientific 1993

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut, Ludwig-Maximilians-Universität München, Theresienstrasse 39, W-8000, München 2, Germany

    Hans-Otto Georgii

  2. Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse, W-4800, Bielefeld 1, Germany

    Hans Zessin

Authors
  1. Hans-Otto Georgii
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  2. Hans Zessin
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Georgii, HO., Zessin, H. Large deviations and the maximum entropy principle for marked point random fields. Probab. Th. Rel. Fields 96, 177–204 (1993). https://doi.org/10.1007/BF01192132

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  • Received: 05 March 1992

  • Revised: 23 December 1992

  • Issue Date: June 1993

  • DOI: https://doi.org/10.1007/BF01192132

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Keywords

  • Entropy
  • Stochastic Process
  • Probability Theory
  • Poisson Distribution
  • Random Field
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