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Lattice trees and super-Brownian motion

Published online by Cambridge University Press:  20 November 2018

Eric Derbez  and
Gordon Slade
Eric Derbez
Affiliation:
Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 e-mail: [email protected]@math.ubc.ca
Gordon Slade
Affiliation:
Department of Mathematics and Statistics McMaster University Hamilton, Ontario L8S 4K1 e-mail: [email protected]@math.ubc.ca
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Abstract

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This article discusses our recent proof that above eight dimensions the scaling limit of sufficiently spread-out lattice trees is the variant of super-Brownian motion called integrated super-Brownian excursion (ISE), as conjectured by Aldous. The same is true for nearest-neighbour lattice trees in sufficiently high dimensions. The proof, whose details will appear elsewhere, uses the lace expansion. Here, a related but simpler analysis is applied to show that the scaling limit of a mean-field theory is ISE, in all dimensions. A connection is drawn between ISE and certain generating functions and critical exponents, which may be useful for the study of high-dimensional percolation models at the critical point.

Keywords

Primary: 82B4160K3560J65
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 1 , 01 March 1997 , pp. 19 - 38
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Aizenman, M., On the number of incipient spanning clusters, Nucl. Phys. B [FS] 485 (1997), 551–582.Google Scholar
2. Aizenman, M. and Newman, C. M., Tree graph inequalities and critical behavior in percolation models, J. Statist. Phys. 36 (1984), 107–143.Google Scholar
3. Aldous, D., The continuum random tree. I, Ann. Probab. 19 (1991), 1–28.Google Scholar
4. Aldous, D., The continuum random tree II: an overview. In: Stochastic Analysis, (eds. M. T. Barlow and N. H. Bingham), Cambridge Univ. Press, Cambridge, 1991, 23–70.Google Scholar
5. Aldous, D., The continuum random tree III, Ann. Probab. 21 (1993), 248–289.Google Scholar
6. Aldous, D., Tree-based models for random distribution of mass, J. Statist. Phys. 73 (1993), 625–641.Google Scholar
7. Bovier, A., J. Fröhlich, and Glaus, U., Branched polymers and dimensional reduction. In: Critical Phenomena, Random Systems, Gauge Theories, (eds. K. Osterwalder and R. Stora), Amsterdam, 1986, North-Holland, Les Houches, 1984.Google Scholar
8. Brydges, D.C. and Spencer, T., Self-avoiding walk in 5 or more dimensions, Comm.Math. Phys. 97 (1985), 125–148.Google Scholar
9. Dawson, D. and Perkins, E., Measure-valued processes and renormalization of branching particle systems. In: Stochastic partial differential equations: Six perspectives, (eds. R. Carmona and B. Rozovskii), AMS Math. Surveys and Monographs, 1997.Google Scholar
10. Derbez, E., The scaling limit of lattice trees above eight dimensions, PhD thesis, McMaster University, (1996).Google Scholar
11. Derbez, E. and Slade, G., The scaling limit of lattice trees in high dimensions, preprint.Google Scholar
12. Goulden, I. P. and Jackson, D. M., Combinatorial Enumeration, John Wiley and Sons, New York, 1983.Google Scholar
13. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press, New York, 4th edition, 1965.Google Scholar
14. Grimmett, G., Percolation, Springer, Berlin, 1989.Google Scholar
15. Hara, T. and Slade, G., Mean-field critical behaviour for percolation in high dimensions, Comm. Math. Phys. 128 (1990), 333–391.Google Scholar
16. Hara, T. and Slade, G., On the upper critical dimension of lattice trees and lattice animals, J. Statist. Phys. 59 (1990), 1469–1510.Google Scholar
17. Hara, T. and Slade, G., The lace expansion for self-avoiding walk in five or more dimensions, Rev.Math. Phys. 4 (1992), 235–327.Google Scholar
18. Hara, T. and Slade, G., The number and size of branched polymers in high dimensions, J. Statist. Phys. 67 (1992), 1009–1038.Google Scholar
19. Hara, T. and Slade, G., Self-avoiding walk in five or more dimensions. I. The critical behaviour, Comm. Math. Phys. 147 (1992), 101–136.Google Scholar
20. Harary, F. and Palmer, E. M., Graphical Enumeration, Academic Press, New York, 1973.Google Scholar
21. van Rensburg, E. J. Janse, On the number of trees in Zd, J. Phys. A, Math. Gen. 25 (1992), 3523–3528.Google Scholar
22. van Rensburg, E. J. Janse and Madras, N., A non-local Monte-Carlo algorithm for lattice trees, J. Phys. A, Math. Gen. 25 (1992), 303–333.Google Scholar
23. Klein, D. J., Rigorous results for branched polymer models with excluded volume, J. Chem. Phys. 75 (1981), 5186–5189.Google Scholar
24. Le Gall, J.-F., The uniform random tree in a Brownian excursion, Probab. Theor. Relat. Fields 96 (1993), 369–383.Google Scholar
25. Lubensky, T. C. and Isaacson, J., Statistics of lattice animals and dilute branched polymers, Phys. Rev. A 20 (1979), 2130–2146.Google Scholar
26. Madras, N., A rigorous bound on the critical exponent for the number of lattice trees, animals and polygons, J. Statist. Phys. 78 (1995), 681–699.Google Scholar
27. Madras, N. and Slade, G., The Self-Avoiding Walk, Birkhäuser, Boston, 1993.Google Scholar
28. Meir, A. and Moon, J. W., The asymptotic behaviour of coefficients of powers of certain generating functions, European J. Combin. 11 (1990), 581–587.Google Scholar
29. Nguyen, B. G. and Yang, W.-S., Triangle condition for oriented percolation in high dimensions, Ann. Probab. 21 (1993), 1809–1844.Google Scholar
30. Nguyen, B. G. and Yang, W.-S., Gaussian limit for critical oriented percolation in high dimensions, J. Statist. Phys. 78 (1995), 841–876.Google Scholar
31. Tasaki, H. and Hara, T., Critical behaviour in a systemof branched polymers, Progr. Theoret. Phys. Suppl. 92 (1987), 14–25.Google Scholar