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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
Published electronically: July 31, 1998
MathSciNet review: 1637050
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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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  • 1. M. Aizenman, On the number of incipient spanning clusters, Nucl. Phys. B [FS] 485 (1997), 551-582. CMP 97:07
  • 2. 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
  • 3. 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,
  • 4. David Aldous, The continuum random tree. III, Ann. Probab. 21 (1993), no. 1, 248–289. MR 1207226
  • 5. David Aldous, Tree-based models for random distribution of mass, J. Statist. Phys. 73 (1993), no. 3-4, 625–641. MR 1251658,
  • 6. D. J. Barsky and M. Aizenman, Percolation critical exponents under the triangle condition, Ann. Probab. 19 (1991), no. 4, 1520–1536. MR 1127713
  • 7. C. Borgs, J.T. Chayes, H. Kesten, and J. Spencer, The birth of the infinite cluster: finite size scaling in percolation. In preparation.
  • 8. -, Uniform boundedness of critical crossing probabilities implies hyperscaling. In preparation.
  • 9. 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
  • 10. 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.
  • 11. E. Derbez and G. Slade, Lattice trees and super-Brownian motion, Canad. Math. Bull. 40 (1997), 19-38. CMP 97:11
  • 12. -, The scaling limit of lattice trees in high dimensions, Commun. Math. Phys. 193 (1998), 69-104. CMP 98:11
  • 13. Geoffrey Grimmett, Percolation, Springer-Verlag, New York, 1989. MR 995460
  • 14. -, Percolation and disordered systems (St. Flour lecture notes, 1996), Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997. CMP 98:06
  • 15. T. Hara and G. Slade, The scaling limit of the incipient infinite cluster in high-dimensional percolation. I. Critical exponents. In preparation.
  • 16. -, The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion. In preparation.
  • 17. 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
  • 18. 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,
  • 19. 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
  • 20. 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
  • 21. Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkhäuser, Boston, Mass., 1982. MR 692943
  • 22. Harry Kesten, The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields 73 (1986), no. 3, 369–394. MR 859839,
  • 23. Jean-François Le Gall, The uniform random tree in a Brownian excursion, Probab. Theory Related Fields 96 (1993), no. 3, 369–383. MR 1231930,
  • 24. Neal Madras and Gordon Slade, The self-avoiding walk, Probability and its Applications, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1197356

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Additional Information

Takashi Hara
Affiliation: Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, Meguro-ku, Tokyo 152, Japan

Gordon Slade
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

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