The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents

Journal of Statistical Physics

Volumn 99, Issue 5-6, 2000, Pages 1075-1168

  Hara, Takashi ^a,b   Slade, Gordon ^c,d  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^b NAGOYA UNIVERSITY   (Japan)

^c MCMASTER UNIVERSITY   (Canada)

^d UNIVERSITY OF BRITISH COLUMBIA   (Canada)

Author keywords

Critical exponents; Incipient infinite cluster; Lace expansion; Percolation

Indexed keywords

EID: 0034336207     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1018628503898     Document Type: Article

References (32)

1. M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36:107-143 (1984).
2. D. J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition, Ann. Probab. 19:1520-1536 (1991).
3. B. G. Nguyen, Gap exponents tor percolation processes with triangle condition, J. Stat. Phys. 49:235-243 (1987).
4. W-S. Yang and Y. Zhang, A note on differentiability of the cluster density for independent percolation in high dimensions, J. Stat. Phys. 66:1123-1138 (1992).
5. T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128:333-391 (1990).
6. T. Hara and G. Slade, Mean-field behaviour and the lace expansion, in Probability and Phase Transition, G. Grimmett, ed. (Dordrecht, Kluwer, 1994).
7. T. Hara, Mean field critical behaviour for correlation length for percolation in high dimensions, Prob. Th. and Rel. Fields 86:337-385 (1990).
8. D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys. 73:625-641 (1993).
9. C. Borgs, J. T. Chayes, R. van der Hotstad, and G. Slade, Mean-field lattice trees, Ann. Combinatories 3:205-221 (1999).
10. J.-F. Le Gall, Branching processes, random trees and superprocesses, in Proceedings of the International Congress of Mathematicians, Berlin 1998, Vol. III, pp. 279-289 (1998). Documenta Mathematica, Extra Volume ICM 1998.
11. T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion, J. Math. Phys. 41:1244-1293 (2000).
12. T. Hara and G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys. 67:1009-1038 (1992).
13. E. Derbez and G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull. 40:19-38 (1997).
14. E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys. 193:69-104 (1998).
15. T. Hara and G. Slade, The incipient infinite cluster in high-dimensional percolation, Electron. Res. Announc. Amer. Math. Soc. 4:48-55 (1998). http://www.ams.org/era/.
16. 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).
17. G. Grimmett, Percolation (Springer, Berlin, 1989).
18. J. T. Chayes, L. Chayes, and R. Durrett, Inhomogeneous percolation problems and incipient infinite clusters, J. Phys. A: Math. Gen. 20:1521-1530 (1987).
19. H. Kesten, The incipient infinite cluster in two-dimensional percolation, Probab. Th. Rel. Fields 73:369-394 (1986).
20. M. Aizenman, The geometry of critical percolation and conformal invariance, in STATPHYS 19, Proceedings Xiamen 1995, Hao Bai-Lin, ed. (World Scientific, 1996).
21. M. Aizenman, On the number of incipient spanning clusters, Nucl. Phys. B [FS] 485:551-582 (1997).
22. M. Aizenman, Scaling limit for the incipient spanning clusters, in Mathematics of Materials: Percolation and Composites. K. M. Golden, G. R. Grimmett, R. D. James, G. W. Milton, and P. N. Sen, eds. (Springer, 1997).
23. M. Aizenman and A. Burchard, Hölder regularity and dimension bounds for random curves. Duke Math. J. 99:419-453 (1999).
24. J.-F. Le Gall, The uniform random tree in a Brownian excursion, Probab. Th. Rel. Fields 96:369-383 (1993).
25. J.-F. Le Gall, Spatial Branching Processes, Random Snakes, and Partial Differential Equations (Birkhäuser, Basel, 1999).
26. G. Toulouse, Perspectives from the theory of phase transitions, Nuoro Cimento B 23:234-240 (1974).
27. 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 (1998).
28. 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).
29. T. Reisz, A convergence theorem for lattice Feynman integrals with massless propagators, Commun. Math. Phys. 116:573-606 (1988).
30. T. Reisz, A power counting theorem for Feynman integrals on the lattice, Commun. Math. Phys. 116:81-126 (1988).
31. N. Madras and G. Slade, The Self-Avoiding Walk (Birkhäuser, Boston, 1993).
32. T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions, Combinatorics, Probability and Computing 4:197-215 (1995).