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Phase Transition and Percolation in Gibbsian Particle Models

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Statistical Physics and Spatial Statistics

Part of the book series: Lecture Notes in Physics ((LNP,volume 554))

Abstract

We discuss the interrelation between phase transitions in interacting lattice or continuum models, and the existence of infinite clusters in suitable random-graph models. In particular, we describe a random-geometric approach to the phase transition in the continuum Ising model of two species of particles with soft or hard interspecies repulsion. We comment also on the related area-interaction process and on perfect simulation.

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References

  1. Baddeley, A.J., M.N.M. van Lieshout (1995): ‘Area-interaction point processes’, Ann. Inst. Statist. Math. 46, pp. 601–619

    Article  Google Scholar 

  2. Chayes, J.T., L. Chayes, R. Kotecký (1995): ‘The analysis of the Widom-Rowlinson model by stochastic geometric methods’, Commun. Math. Phys. 172, pp. 551–569

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Edwards, R.G., A.D. Sokal (1988): ‘Generalization of the Fortuin-Kasteleyn—Swendsen-Wang representation and Monte Carlo algorithm’, Phys. Rev. D 38, pp. 2009–2012

    ADS  MathSciNet  Google Scholar 

  4. Fortuin, C.M., P.W. Kasteleyn (1972): ‘On the random-cluster model. I. Introduction and relation to other models’, Physica 57, pp. 536–564

    Article  ADS  MathSciNet  Google Scholar 

  5. Georgii, H.-O. (1988): Gibbs Measures and Phase Transitions (de Gruyter, Berlin New York)

    MATH  Google Scholar 

  6. Georgii, H.-O. (1994): ‘Large Deviations and the Equivalence of Ensembles for Gibbsian Particle Systems with Superstable Interaction’, Probab. Th. Rel. Fields 99, pp. 171–195

    Article  MATH  MathSciNet  Google Scholar 

  7. Georgii, H.-O. (1995): ‘The Equivalence of Ensembles for Classical Systems of Particles’, J. Statist. Phys. 80, pp. 1341–1378

    Article  MATH  MathSciNet  ADS  Google Scholar 

  8. Georgii, H.-O. (1999): ‘Translation invariance and continuous symmetries in two dimensional continuum systems’. In: Mathematical results in Statistical Mechanics, ed. by S. Miracle-Sole, J. Ruiz, V. Zagrebnov (World Scientific, Singapore etc.), pp. 53–69

    Google Scholar 

  9. Georgii, H.-O., O. Häggström (1996): ‘Phase transition in continuum Potts models’, Commun. Math. Phys. 181, pp. 507–528

    Article  MATH  ADS  Google Scholar 

  10. Georgii, H.-O., O. Häggström, C. Maes (1999): ‘The random geometry of equilibrium phases’. In: Critical phenomena, ed. by Domb and J.L. Lebowitz (Academic Press), forthcoming

    Google Scholar 

  11. Georgii, H.-O., Y. Higuchi (1999): ‘Percolation and number of phases in the 2D Ising model’, submitted to J. Math. Phys.

    Google Scholar 

  12. Georgii, H.-O., T. Küneth (1997): ‘Stochastic comparison of point random fields’, J. Appl. Probab. 34, pp. 868–881

    Article  MATH  MathSciNet  Google Scholar 

  13. Grimmett, G.R. (1999): Percolation, 2nd ed. (Springer, New York)

    MATH  Google Scholar 

  14. Gruber, Ch., R.B. Griffiths (1986): ‘Phase transition in a ferromagnetic fluid’, Physica A 138, pp. 220–230

    Article  ADS  MathSciNet  Google Scholar 

  15. Häggström, O., M.N.M. van Lieshout, J. Moller (1997): ‘Characterization results and Markov chain Monte Carlo algorithms including exact simulation for some spatial point processes’, Bernoulli, to appear

    Google Scholar 

  16. Likos, C.N, K.R. Mecke, H. Wagner (1995): ‘Statistical morphology of random interfaces in microemulsions’, J. Chem. Phys. 102, pp. 9350–9361

    Article  ADS  Google Scholar 

  17. Mecke, K.R. (1996): ‘A morphological model for complex fluids’, J. Phys.: Condens. Matter 8, pp. 9663–9668

    Article  ADS  Google Scholar 

  18. Meester, R., R. Roy (1996): Continuum Percolation (Cambridge University Press)

    Google Scholar 

  19. Penrose, M.D. (1991): ‘On a continuum percolation model’, Adv. Appl. Probab. 23, pp. 536–556

    Article  MATH  MathSciNet  Google Scholar 

  20. Swendsen, R.H., J.-S. Wang (1987): ‘Nonuniversal critical dynamics in Monte Carlo simulations’, Phys. Rev. Lett. 58, pp. 86–88

    Article  ADS  Google Scholar 

  21. Thönnes, E. (1999): ‘Perfect simulation of some point processes for the impatient user’, Adv. Appl. Probab. 31, pp. 69–87

    Article  MATH  Google Scholar 

  22. Widom, B., J.S. Rowlinson (1970): ‘New model for the study of liquid-vapor phase transition’, J. Chem. Phys. 52, pp. 1670–1684

    Article  ADS  Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematisches Institut der Universität München, Theresienstr. 39, D-80333, München, Germany

    Hans-Otto Georgii

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  1. Hans-Otto Georgii
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Editor information

Editors and Affiliations

  1. Fachbereich Physik, Bergische Universität Wuppertal, 42097, Wuppertal, Germany

    Klaus R. Mecke

  2. Institut für Stochastik, TU Bergakademie Freiberg, 09596, Freiberg, Germany

    Dietrich Stoyan

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© 2000 Springer-Verlag Berlin Heidelberg

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Georgii, HO. (2000). Phase Transition and Percolation in Gibbsian Particle Models. In: Mecke, K.R., Stoyan, D. (eds) Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol 554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45043-2_10

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  • DOI: https://doi.org/10.1007/3-540-45043-2_10

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67750-5

  • Online ISBN: 978-3-540-45043-6

  • eBook Packages: Springer Book Archive

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