The incipient infinite cluster in high-dimensional percolation
Authors:
Takashi Hara and Gordon Slade
Journal:
Electron. Res. Announc. Amer. Math. Soc. 4 (1998), 48-55
MSC (1991):
Primary 82B43, 60K35
DOI:
https://doi.org/10.1090/S1079-6762-98-00046-8
Published electronically:
July 31, 1998
MathSciNet review:
1637050
Full-text PDF Free Access
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Abstract: We announce our recent proof that, for independent bond percolation in high dimensions, the scaling limits of the incipient infinite cluster’s two-point and three-point functions are those of integrated super-Brownian excursion (ISE). The proof uses an extension of the lace expansion for percolation.
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- M. Aizenman and C.M. Newman, Tree graph inequalities and critical behavior in percolation models, J. Stat. Phys. 36 (1984), 107–143.
- D. Aldous, The continuum random tree III, Ann. Probab. 21 (1993), 248–289.
- ---, Tree-based models for random distribution of mass, J. Stat. Phys. 73 (1993), 625–641.
- D.J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition, Ann. Probab. 19 (1991), 1520–1536.
- C. Borgs, J.T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster: finite size scaling in percolation. In preparation.
- ---, Uniform boundedness of critical crossing probabilities implies hyperscaling. In preparation.
- J.T. Chayes, L. Chayes, and R. Durrett, Inhomogeneous percolation problems and incipient infinite clusters, J. Phys. A: Math. Gen. 20 (1987), 1521–1530.
- D. Dawson and E. Perkins, Measure-valued processes and renormalization of branching particle systems, Stochastic Partial Differential Equations: Six Perspectives (R. Carmona and B. Rozovskii, eds.), AMS Math. Surveys and Monographs. To appear.
- E. Derbez and G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull. 40 (1997), 19–38.
- ---, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys. 193 (1998), 69–104.
- G. Grimmett, Percolation, Springer, Berlin, 1989.
- ---, Percolation and disordered systems (St. Flour lecture notes, 1996), Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997.
- T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. In preparation.
- ---, The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. In preparation.
- ---, Mean-field critical behaviour for percolation in high dimensions, Commun. Math. Phys. 128 (1990), 333–391.
- ---, The number and size of branched polymers in high dimensions, J. Stat. Phys. 67 (1992), 1009–1038.
- ---, Mean-field behaviour and the lace expansion, Probability and Phase Transition (Dordrecht) (G. Grimmett, ed.), Kluwer, 1994.
- B.D. Hughes, Random walks and random environments, vol. 2: Random Environments, Oxford University Press, New York, 1996.
- H. Kesten, Percolation theory for mathematicians, Birkhäuser, Boston, 1982.
- ---, The incipient infinite cluster in two-dimensional percolation, Probab. Th. Rel. Fields 73 (1986), 369–394.
- J.-F. Le Gall, The uniform random tree in a Brownian excursion, Probab. Th. Rel. Fields 96 (1993), 369–383.
- N. Madras and G. Slade, The self-avoiding walk, Birkhäuser, Boston, 1993.
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Additional Information
Takashi Hara
Affiliation:
Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan
Email:
hara@ap.titech.ac.jp
Gordon Slade
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1
Email:
slade@mcmaster.ca
Keywords:
Critical exponent,
incipient infinite cluster,
integrated super-Brownian excursion,
percolation,
scaling limit,
super-Brownian motion
Received by editor(s):
March 17, 1998
Received by editor(s) in revised form:
May 20, 1998
Published electronically:
July 31, 1998
Communicated by:
Klaus Schmidt
Article copyright:
© Copyright 1998
American Mathematical Society
