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

Abstract |
References |
Similar Articles |
Additional Information

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.

- M. Aizenman,
*On the number of incipient spanning clusters*, Nucl. Phys. B [FS] ** 485** (1997), 551–582.
- Michael Aizenman and David J. Barsky,
*Sharpness of the phase transition in percolation models*, Comm. Math. Phys. **108** (1987), no. 3, 489–526. MR **874906**, DOI 10.1007/BF01212322
- Michael Aizenman and Charles M. Newman,
*Tree graph inequalities and critical behavior in percolation models*, J. Statist. Phys. **36** (1984), no. 1-2, 107–143. MR **762034**, DOI 10.1007/BF01015729
- David Aldous,
*The continuum random tree. III*, Ann. Probab. **21** (1993), no. 1, 248–289. MR **1207226**
- David Aldous,
*Tree-based models for random distribution of mass*, J. Statist. Phys. **73** (1993), no. 3-4, 625–641. MR **1251658**, DOI 10.1007/BF01054343
- D. J. Barsky and M. Aizenman,
*Percolation critical exponents under the triangle condition*, Ann. Probab. **19** (1991), no. 4, 1520–1536. MR **1127713**, DOI 10.1214/aop/1176990221
- 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 **20** (1987), no. 6, 1521–1530. MR **893330**, DOI 10.1088/0305-4470/20/6/034
- 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.
- Geoffrey Grimmett,
*Percolation*, Springer-Verlag, New York, 1989. MR **995460**, DOI 10.1007/978-1-4757-4208-4
- —,
*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.
- Takashi Hara and Gordon Slade,
*Mean-field critical behaviour for percolation in high dimensions*, Comm. Math. Phys. **128** (1990), no. 2, 333–391. MR **1043524**, DOI 10.1007/BF02108785
- Takashi Hara and Gordon Slade,
*The number and size of branched polymers in high dimensions*, J. Statist. Phys. **67** (1992), no. 5-6, 1009–1038. MR **1170084**, DOI 10.1007/BF01049008
- Takashi Hara and Gordon Slade,
*Mean-field behaviour and the lace expansion*, Probability and phase transition (Cambridge, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 420, Kluwer Acad. Publ., Dordrecht, 1994, pp. 87–122. MR **1283177**
- Barry D. Hughes,
*Random walks and random environments. Vol. 2*, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. Random environments. MR **1420619**
- Harry Kesten,
*Percolation theory for mathematicians*, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982. MR **692943**, DOI 10.1007/978-1-4899-2730-9
- Harry Kesten,
*The incipient infinite cluster in two-dimensional percolation*, Probab. Theory Related Fields **73** (1986), no. 3, 369–394. MR **859839**, DOI 10.1007/BF00776239
- Jean-François Le Gall,
*The uniform random tree in a Brownian excursion*, Probab. Theory Related Fields **96** (1993), no. 3, 369–383. MR **1231930**, DOI 10.1007/BF01292678
- Neal Madras and Gordon Slade,
*The self-avoiding walk*, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR **1197356**

- M. Aizenman,
*On the number of incipient spanning clusters*, Nucl. Phys. B [FS] ** 485** (1997), 551–582.
- M. Aizenman and D.J. Barsky,
*Sharpness of the phase transition in percolation models*, Commun. Math. Phys. ** 108** (1987), 489–526.
- 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.

Similar Articles

Retrieve articles in *Electronic Research Announcements of the American Mathematical Society*
with MSC (1991):
82B43,
60K35

Retrieve articles in all journals
with MSC (1991):
82B43,
60K35

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