Lattice trees and super-Brownian motion

Derbez, Eric ; Slade, Gordon

Canadian mathematical bulletin, Tome 40 (1997) no. 1, pp. 19-38

Résumé

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.

Détail

DOI : 10.4153/CMB-1997-003-8

Mots-clés : Primary: 82B41, 60K35, 60J65

Bibliographie

[1] 1. Aizenman, M., On the number of incipient spanning clusters, Nucl. Phys. B [FS] 485 (1997), 551–582. Google Scholar

[2] 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] 3. Aldous, D., The continuum random tree. I, Ann. Probab. 19 (1991), 1–28. Google Scholar

[4] 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] 5. Aldous, D., The continuum random tree III, Ann. Probab. 21 (1993), 248–289. Google Scholar

[6] 6. Aldous, D., Tree-based models for random distribution of mass, J. Statist. Phys. 73 (1993), 625–641. Google Scholar

[7] 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] 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] 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] 10. Derbez, E., The scaling limit of lattice trees above eight dimensions, PhD thesis, McMaster University, (1996). Google Scholar

[11] 11. Derbez, E. and Slade, G., The scaling limit of lattice trees in high dimensions, preprint. Google Scholar

[12] 12. Goulden, I. P. and Jackson, D. M., Combinatorial Enumeration, John Wiley and Sons, New York, 1983. Google Scholar

[13] 13. Gradshteyn, I. S. and Ryzhik, I. M., Table of Integrals, Series and Products, Academic Press, New York, 4th edition, 1965. Google Scholar

[14] 14. Grimmett, G., Percolation, Springer, Berlin, 1989. Google Scholar

[15] 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] 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] 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] 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] 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] 20. Harary, F. and Palmer, E. M., Graphical Enumeration, Academic Press, New York, 1973. Google Scholar

[21] 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] 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] 23. Klein, D. J., Rigorous results for branched polymer models with excluded volume, J. Chem. Phys. 75 (1981), 5186–5189. Google Scholar

[24] 24. Le Gall, J.-F., The uniform random tree in a Brownian excursion, Probab. Theor. Relat. Fields 96 (1993), 369–383. Google Scholar

[25] 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] 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] 27. Madras, N. and Slade, G., The Self-Avoiding Walk, Birkhäuser, Boston, 1993. Google Scholar

[28] 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] 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] 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] 31. Tasaki, H. and Hara, T., Critical behaviour in a systemof branched polymers, Progr. Theoret. Phys. Suppl. 92 (1987), 14–25. Google Scholar