Abstract
This is the first of two papers on the critical behavior of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents η and δ for the nearest neighbor model in very high dimensions d≫6 and for sufficiently spread-out models in all dimensions d>6. The exponent η describes the low-frequency behavior of the Fourier transform of the critical two-point connectivity function, while δ describes the behavior of the magnetization at the critical point. Our main result is an asymptotic relation showing that, in a joint sense, η=0 and δ=2. The proof uses a major extension of our earlier expansion method for percolation. This result provides evidence that the scaling limit of the incipient infinite cluster is the random probability measure on ℝd known as integrated super-Brownian excursion (ISE), in dimensions above 6. In the sequel to this paper, we extend our methods to prove that the scaling limits of the incipient infinite cluster's two-point and three-point functions are those of ISE for the nearest neighbor model in dimensions d≫6.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
REFERENCES
M. Aizenman and C. M. Newman, Tree graph inequalities and critical behavior in per-colation models, J. Stat. Phys. 36:107–143 (1984).
D. J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condi-tion, Ann. Probab. 19:1520–1536 (1991).
B. G. Nguyen, Gap exponents for percolation processes with triangle condition, J. Stat. Phys. 49:235–243 (1987).
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).
T. Hara and G. Slade, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128:333–391 (1990).
T. Hara and G. Slade, Mean-field behaviour and the lace expansion, in Probability and Phase Transition, G. Grimmett, ed. (Dordrecht, Kluwer, 1994).
T. Hara, Mean field critical behaviour for correlation length for percolation in high dimensions, Prob. Th. and Rel. Fields 86:337–385 (1990).
D. Aldous, Tree-based models for random distribution of mass, J. Stat. Phys. 73:625–641 (1993).
C. Borgs, J. T. Chayes, R. van der Hofstad, and G. Slade, Mean-field lattice trees, Ann. Combinatorics 3:205–221 (1999).
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.
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).
T. Hara and G. Slade, The number and size of branched polymers in high dimensions, J. Stat. Phys. 67:1009–1038 (1992).
E. Derbez and G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull. 40:19–38 (1997).
E. Derbez and G. Slade, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys. 193:69–104 (1998).
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/.
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. (Birkhä user, Basel, 1999).
G. Grimmett, Percolation (Springer, Berlin, 1989).
J. T. Chayes, L. Chayes, and R. Durrett, Inhomogeneous percolation problems and incipient infinite clusters, J. Phys. A: Math. Gen. 20:1521–1530 (1987).
H. Kesten, The incipient infinite cluster in two-dimensional percolation, Probab. Th. Rel. Fields 73:369–394 (1986).
M. Aizenman, The geometry of critical percolation and conformal invariance, in STATPHYS 19, Proceedings Xiamen 1995, Hao Bai-Lin, ed. (World Scientific, 1996).
M. Aizenman, On the number of incipient spanning clusters, Nucl. Phys. B [FS] 485:551–582 (1997)
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).
M. Aizenman and A. Burchard, Hö lder regularity and dimension bounds for random curves, Duke Math. J. 99:419–453 (1999).
J.-F. Le Gall, The uniform random tree in a Brownian excursion, Probab. Th. Rel. Fields 96:369–383 (1993).
J.-F. Le Gall, Spatial Branching Processes, Random Snakes, and Partial Differential Equations (Birkhä user, Basel, 1999).
G. Toulouse, Perspectives from the theory of phase transitions, Nuovo Cimento B 23:234–240 (1974).
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).
J.-F. Le Gall, The Hausdorff measure of the range of super-Brownian motion, in Perplex-ing Problems in Probability: Festschrift in Honor of Harry Kesten, M. Bramson and R. Durrett, eds. (Birkhä user, Basel, 1999).
T. Reisz, A convergence theorem for lattice Feynman integrals with massless propagators, Commun. Math. Phys. 116:573–606 (1988).
T. Reisz, A power counting theorem for Feynman integrals on the lattice, Commun. Math. Phys. 116:81–126 (1988).
N. Madras and G. Slade, The Self-Avoiding Walk (Birkhä user, Boston, 1993).
T. Hara and G. Slade, The self-avoiding-walk and percolation critical points in high dimensions, Combinatorics, Probability and Computing 4:197–215 (1995).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Hara, T., Slade, G. The Scaling Limit of the Incipient Infinite Cluster in High-Dimensional Percolation. I. Critical Exponents. Journal of Statistical Physics 99, 1075–1168 (2000). https://doi.org/10.1023/A:1018628503898
Issue Date:
DOI: https://doi.org/10.1023/A:1018628503898