The Annals of Probability

Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

Takashi Hara, Gordon Slade, and Remco van der Hofstad

Full-text: Open access


We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ${\mathbb{Z}^d}$, having long finite-range connections, above their upper critical dimensions $d=4$ (self-avoiding walk), $d=6$ (percolation) and $d=8$ (trees and animals). The two-point functions for these models are respectively the generating function for self-avoiding walks from the origin to $x \in {\mathbb{Z}^d}$, the probability of a connection from 0 to x, and the generating function for lattice trees or lattice animals containing 0 and x. We use the lace expansion to prove that for sufficiently spread-out models above the upper critical dimension, the two-point function of each model decays, at the critical point, as a multiple of $|x|^{2-d}$ as $x \to \infty$. We use a new unified method to prove convergence of the lace expansion. The method is based on x-space methods rather than the Fourier transform. Our results also yield unified and simplified proofs of the bubble condition for self-avoiding walk, the triangle condition for percolation, and the square condition for lattice trees and lattice animals, for sufficiently spread-out models above the upper critical dimension.

Article information

Ann. Probab., Volume 31, Number 1 (2003), 349-408.

First available in Project Euclid: 26 February 2003

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Critical exponent lace expansion lattice tree lattice animal percolation self-avoiding walk


Hara, Takashi; van der Hofstad, Remco; Slade, Gordon. Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Probab. 31 (2003), no. 1, 349--408. doi:10.1214/aop/1046294314.

Export citation


  • ADLER, J., MEIR, Y., AHARONY, A. and HARRIS, A. B. (1990). Series study of percolation moments in general dimension. Phy s. Rev. B 41 9183-9206.
  • AIZENMAN, M. (1997). On the number of incipient spanning clusters. Nucl. Phy s. B 485 551-582.
  • AIZENMAN, M. and BARSKY, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phy s. 108 489-526.
  • AIZENMAN, M., KESTEN, H. and NEWMAN, C. M. (1987). Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation. Comm. Math. Phy s. 111 505-531.
  • AIZENMAN, M. and NEWMAN, C. M. (1984). Tree graph inequalities and critical behavior in percolation models. J. Statist. Phy s. 36 107-143.
  • BARSKY, D. J. and AIZENMAN, M. (1991). Percolation critical exponents under the triangle condition. Ann. Probab. 19 1520-1536.
  • BOLTHAUSEN, E. and RITZMANN, C. (2001). Strong pointwise estimates for the weakly selfavoiding walk. Ann. Probab. To appear.
  • BOVIER, A., FRÖHLICH, J. and GLAUS, U. (1986). Branched poly mers and dimensional reduction. In Critical Phenomena, Random Sy stems, Gauge Theories (K. Osterwalder and R. Stora, eds.) 850-893. North-Holland, Amsterdam.
  • BRy DGES, D., EVANS, S. N. and IMBRIE, J. Z. (1992). Self-avoiding walk on a hierarchical lattice in four dimensions. Ann. Probab. 20 82-124.
  • BRy DGES, D. C. and SPENCER, T. (1985). Self-avoiding walk in 5 or more dimensions. Comm. Math. Phy s. 97 125-148.
  • DERBEZ, E. and SLADE, G. (1998). The scaling limit of lattice trees in high dimensions. Comm. Math. Phy s. 193 69-104.
  • GRIMMETT, G. (1999). Percolation, 2nd ed. Springer, Berlin.
  • HAMMERSLEY, J. M. and MORTON, K. W. (1954). Poor man's Monte Carlo. J. Roy. Statist. Soc. Ser. B 16 23-38.
  • HARA, T. (2003). Critical two-point functions for nearest-neighbour high-dimensional self-avoiding walk and percolation. Unpublished manuscript.
  • HARA, T. and SLADE, G. (1990a). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phy s. 128 333-391.
  • HARA, T. and SLADE, G. (1990b). On the upper critical dimension of lattice trees and lattice animals. J. Statist. Phy s. 59 1469-1510.
  • HARA, T. and SLADE, G. (1994). Mean-field behaviour and the lace expansion. In Probability and Phase Transition (G. Grimmett, ed.) 87-122. Kluwer, Dordrecht.
  • HARA, T. and SLADE, G. (1995). The self-avoiding-walk and percolation critical points in high dimensions. Combin. Probab. Comput. 4 197-215.
  • HARA, T. and SLADE, G. (2000). The scaling limit of the incipient infinite cluster in highdimensional percolation. II. Integrated super-Brownian excursion. J. Math. Phy s. 41 1244-1293.
  • VAN DER HOFSTAD, R. and SLADE, G. (2002a). A generalised inductive approach to the lace expansion. Probab. Theory Related Fields 122 389-430.
  • VAN DER HOFSTAD, R. and SLADE, G. (2002b). The lace expansion on a tree with application to networks of self-avoiding walks. Adv. in Appl. Math. To appear.
  • HUGHES, B. D. (1995). Random Walks and Random Environments 1. Random Walks. Oxford Univ. Press.
  • HUGHES B. D. (1996). Random Walks and Random Environments 2. Random Environments. Oxford Univ. Press.
  • IAGOLNITZER, D. and MAGNEN, J. (1994). Poly mers in a weak random potential in dimension four: Rigorous renormalization group analysis. Comm. Math. Phy s. 162 85-121.
  • KESTEN, H. (1982). Percolation Theory for Mathematicians. Birkhäuser, Boston.
  • KLARNER, D. A. (1967). Cell growth problems. Canad. J. Math. 19 851-863.
  • KLEIN, D. J. (1981). Rigorous results for branched poly mer models with excluded volume. J. Chem. Phy s. 75 5186-5189.
  • LAWLER, G. F., SCHRAMM, O. and WERNER, W. (2001). One-arm exponent for critical 2D percolation. Electron. J. Probab. 7.
  • LUBENSKY, T. C. and ISAACSON, J. (1979). Statistics of lattice animals and dilute branched poly mers. Phy s. Rev. A 20 2130-2146.
  • MADRAS, N. and SLADE, G. (1993). The Self-Avoiding Walk. Birkhäuser, Boston.
  • MENSHIKOV, M. V. (1986). Coincidence of critical points in percolation problems. Soviet Math. Dokl. 33 856-859.
  • NGUy EN, B. G. and YANG, W.-S. (1993). Triangle condition for oriented percolation in high dimensions. Ann. Probab. 21 1809-1844.
  • PARISI, G. and SOURLAS, N. (1981). Critical behavior of branched poly mers and the Lee-Yang edge singularity. Phy s. Rev. Lett. 46 871-874.
  • PENROSE, M. D. (1994). Self-avoiding walks and trees in spread-out lattices. J. Statist. Phy s. 77 3-15.
  • SPITZER, F. (1976). Principles of Random Walk, 2nd ed. Springer, New York.
  • TASAKI, H. and HARA, T. (1987). Critical behaviour in a sy stem of branched poly mers. Progr. Theoret. Phy s. Suppl. 92 14-25.
  • UCHIy AMA, K. (1998). Green's functions for random walks on ZN. Proc. London Math. Soc. 77 215-240.