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Annals of Combinatorics

, Volume 3, Issue 2–4, pp 205–221 | Cite as

Mean-field lattice trees

  • Christian Borgs
  • Jennifer Chayes
  • Remco van der Hofstad
  • Gordon Slade
Article

Abstract

We introduce a mean-field model of lattice trees based on embeddings into ℤ d of abstract trees having a critical Poisson offspring distribution. This model provides a combinatorial interpretation for the self-consistent mean-field model introduced previously by Derbez and Slade [9], and provides an alternative approach to work of Aldous. The scaling limit of the meanfield model is integrated super-Brownian excursion (ISE), in all dimensions. We also introduce a model of weakly self-avoiding lattice trees, in which an embedded tree receives a penaltye−β for each self-intersection. The weakly self-avoiding lattice trees provide a natural interpolation between the mean-field model (β=0), and the usual model of strictly self-avoiding lattice tress (β=∞) which associates the uniform measure to the set of lattice trees of the same size.

AMS Subject Classification

60K35 82B41 

Keywords

lattice tree mean field scaling limit integrated super-Brownian motion 

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References

  1. 1.
    D. Aldous, The continuum random tree III, Ann. Prob.21 (1993) 248–289.Google Scholar
  2. 2.
    D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys.73 (1993) 625–641.Google Scholar
  3. 3.
    R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.Google Scholar
  4. 4.
    P. Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, 1968.Google Scholar
  5. 5.
    A. Bovier, J. Fröhlich, and U. Glaus, Branched polymers and dimensional reduction, In: Critical Phenomena, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, Eds., North-Holland, Amsterdam, 1986.Google Scholar
  6. 6.
    R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, On the LambertW function, Adv. Comput. Math.5 (1996) 329–359.Google Scholar
  7. 7.
    D. Dawson and E. Perkins, Measure-valued processes and renormalization of branching particle systems, In: Stochastic Partial Differential Equations: Six Perspectives, R. Carmona and B. Rozovskii, Eds., AMS Math. Surveys and Monographs, American Mathematical Society, 1998.Google Scholar
  8. 8.
    D.A. Dawson, Measure-valued Markov processes, In: Ecole d'Eté de Probabilités de Saint-Flour 1991, Lecture Notes in Mathematics, Vol. 1541, Springer-Verlag, Berlin, 1993.Google Scholar
  9. 9.
    E. Derbez and G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull.40 (1997) 19–38.Google Scholar
  10. 10.
    E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys.193 (1998) 69–104.Google Scholar
  11. 11.
    C. Domb and G.S. Joyce, Cluster expansion for a polymer chain, J. Phys. C: Solid State Phys.5 (1972) 956–976.Google Scholar
  12. 12.
    P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math.3 (1990) 216–240.Google Scholar
  13. 13.
    G. Grimmett, Percolation, Springer-Verlag, Berlin, 1989.Google Scholar
  14. 14.
    T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals, J. Stat. Phys.59 (1990) 1469–1510.Google Scholar
  15. 15.
    T. Hara and G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys.67 (1992) 1009–1038.Google Scholar
  16. 16.
    T. Hara and G. Slade, The incipient infinite cluster in high-dimensional percolation, Electron. Res. Announc. Amer. Math. Soc.4 (1998) 48–55; http://www.ams.org/era/.Google Scholar
  17. 17.
    T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation I, Critical exponents, preprint.Google Scholar
  18. 18.
    T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation II, Integrated super-Brownian excursion, preprint.Google Scholar
  19. 19.
    F. Harary and E.M. Palmer, Graphical Enumeration, Academic Press, New York, 1973.Google Scholar
  20. 20.
    J.-F. Le Gall, The uniform random tree in a Brownian excursion, Prob. Th. Rel. Fields96 (1993) 369–383.Google Scholar
  21. 21.
    J.-F. Le Gall, Branching processes, random trees and superprocesses, In: Proceedings of the International Congress of Mathematicians, Berlin 1998, Vol. III, 1998, pp. 279–289; Documenta Mathematica, Extra Volume ICM 1998.Google Scholar
  22. 22.
    J.-F. Le Gall, The Hausdorff measure of the range of super-Brownian motion, In: Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, M. Bramson and R. Durrett, Eds., Birkhäuser, Basel, 1999.Google Scholar
  23. 23.
    J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser, to appear.Google Scholar
  24. 24.
    N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser, Boston, 1993.Google Scholar
  25. 25.
    G. Slade, Lattice trees, percolation and super-Brownian motion, In: Perplexing Problems in Probability: Festschrift in Honor of Harry Kesten, M. Bramson and R. Durrett, Eds., Birkhäuser, Basel, 1999.Google Scholar

Copyright information

© Springer-Verlag 1999

Authors and Affiliations

  • Christian Borgs
    • 1
  • Jennifer Chayes
    • 1
  • Remco van der Hofstad
    • 2
  • Gordon Slade
    • 2
  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada

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