Annals of Combinatorics 3 (1999) 205-221


Mean-Field Lattice Trees

Christian Borgs1, Jennifer Chayes1, Remco van der Hofstad2, and Gordon Slade2

1Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA
{borgs, jchayes}@microsoft.com

2Department of Mathematics and Statistics, McMaster University, Hamilton, OntarioCanada L8S 4K1, Canada

Received October 11, 1998

AMS Subject Classification: 60K35, 82B41

Abstract. We introduce a mean-field model of lattice trees based on embeddings into Zdof 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 mean-field 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 penalty e 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 trees (β =∞) which associates the uniform measure to the set of lattice trees of the same size.

Keywords: lattice tree, mean field, scaling limit, integrated super-Brownian motion


References

1.  D. Aldous, The continuum random tree III, Ann. Prob. 21 (1993) 248–289.

2  D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys. 73 (1993) 625–641.

3.  R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.

4.  P. Billingsley, Convergence of Probability Measures, John Wiley and Sons, New York, 1968.

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.

6.  R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth, On the Lambert W function, Adv. Comput. Math. 5 (1996) 329–359.

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.

8.  D.A. Dawson, Measure-valued Markov processes, In: Ecole dÉtè de Probabilitès de Saint-Flour 1991, Lecture Notes in Mathematics, Vol. 1541, Springer-Verlag, Berlin, 1993.

9.  E. Derbez and G. Slade, Lattice trees and super-{Brownian} motion, Canad. Math. Bull. 40 (1997) 19–38.

10.  E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys. 193 (1998) 69–104.

11.  C. Domb and G.S. Joyce, Cluster expansion for a polymer chain, J. Phys. C: Solid State Phys. 5 (1972) 956–976.

12.  P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM J. Discrete Math. 3 (1990) 216–240.

13.  G. Grimmett, Percolation, Springer-Verlag, Berlin, 1989.

14.  T. Hara and G. Slade, On the upper critical dimension of lattice trees and lattice animals, J. Stat. Phys. 59 (1990) 1469–1510.

15.  T. Hara and G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys. 67 (1992) 1009–1038.

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/.

17.  T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation {I}, {Critical} exponents, preprint.

18.  T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation {II}, Integrated super-Brownian excursion, preprint.

19.  F. Harary and E.M. Palmer, Graphical Enumeration, Academic Press, New York, 1973.

20.  J.-F. Le Gall, The uniform random tree in a {Brownian} excursion, Prob. Th. Rel. Fields 96 (1993) 369–383.

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.

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.

23.  J.-F. Le Gall, Spatial branching processes, random snakes and partial differential equations, Lectures in Mathematics ETH Zürich, Birkhäuser, to appear.

24.  N. Madras and G. Slade, The Self-Avoiding Walk, Birkhäuser, Boston, 1993.

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.


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