The Annals of Probability

Incipient infinite percolation clusters in 2D

Antal A. Járai

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We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC.

We also show that for any $k \ge 1$ the difference in size between the $k$th and $(k+1)$st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.

Article information

Ann. Probab., Volume 31, Number 1 (2003), 444-485.

First available in Project Euclid: 26 February 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

Percolation incipient infinite cluster spanning cluster critical phenomena


Járai, Antal A. Incipient infinite percolation clusters in 2D. Ann. Probab. 31 (2003), no. 1, 444--485. doi:10.1214/aop/1046294317.

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  • AIZENMAN, M. (1997). On the number of incipient spanning clusters. Nuclear Phy s. B 485 551- 582.
  • VAN DEN BERG, J. and KESTEN, H. (1985). Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 556-569.
  • BORGS, C., CHAy ES, J. T., KESTEN, H. and SPENCER, J. (1999). Uniform boundedness of critical crossing probabilities implies hy perscaling. Random Structures Algorithms 15 368-413.
  • BORGS, C., CHAy ES, J. T., KESTEN, H. and SPENCER, J. (2001). Birth of the infinite cluster: Finite-size scaling in percolation. Comm. Math. Phy s. 224 153-204.
  • CHAy ES, J. T. and CHAy ES, L. (1986). Percolation and random media. In Critical Phenomena, Random Sy stem and Gauge Theories (K. Osterwalder and R. Stora, eds.) 1000-1142. North-Holland, Amsterdam.
  • CHAy ES, J. T., CHAy ES, L. and DURRETT, R. (1987). Inhomogeneous percolation problems and incipient infinite clusters. J. Phy s. A 20 1521-1530.
  • CHAy ES, J. T., CHAy ES, L. and FRÖHLICH, J. (1985). The low-temperature behavior of disordered magnets. Comm. Math. Phy s. 100 399-437.
  • DURRETT, R. (1996). Probability: Theory and Examples, 2nd ed. Wadsworth, Belmont, CA.
  • ESSEEN, C. G. (1966). On the Kolmogorov-Rogozin inequality for the concentration function. Z. Warsch. Verw. Gebiete 5 210-216.
  • GRIMMETT, G. R. (1999). Percolation, 2nd ed. Springer, Berlin.
  • HARA, T. and SLADE, G. (2000a). The scaling limit of the incipient infinite cluster in highdimensional percolation. I. Critical exponents. J. Statist. Phy s. 99 1075-1168.
  • HARA, T. and SLADE, G. (2000b). The scaling limit of the incipient infinite cluster in highdimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phy s. 41 1244-1293.
  • JÁRAI, A. A. (2000). Incipient infinite clusters in 2D percolation. Ph.D. dissertation, Cornell Univ. JÁRAI, A. A. Invasion percolation and the incipient infinite cluster in 2D. Unpublished manuscript.
  • KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • KESTEN, H. (1986). The incipient infinite cluster in two-dimensional percolation. Probab. Theory Related Fields 73 369-394.
  • KESTEN, H. (1987). Scaling relations for 2D-percolation. Comm. Math. Phy s. 109 109-156.
  • LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2002). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7.
  • NGUy EN, B. (1988). Ty pical cluster size for two-dimensional percolation processes. J. Statist. Phy s. 50 715-726.
  • RUSSO, L. (1978). A note on percolation. Z. Warsch. Verw. Gebiete 43 39-48.
  • SEy MOUR, P. D. and WELSH, D. J. A. (1978). Percolation probabilities on the square lattice. In Advances in Graph Theory (B. Bollobás, ed.) 227-245. North-Holland, Amsterdam.
  • SMIRNOV, S. (2001). Critical percolation in the plane: Conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 239-244.
  • SMIRNOV, S. and WERNER, W. (2001). Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 729-744.
  • ZHANG, Y. (1999). Some power laws on two dimensional critical bond percolation. Preprint.