The Annals of Probability

Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals

Takashi Hara

Full-text: Open access

Abstract

We consider nearest-neighbor self-avoiding walk, bond percolation, lattice trees, and bond lattice animals on ℤd. The two-point functions of these models are respectively the generating function for self-avoiding walks from the origin to x∈ℤd, the probability of a connection from the origin to x, and the generating functions for lattice trees or lattice animals containing the origin and x. Using the lace expansion, we prove that the two-point function at the critical point is asymptotic to const.|x|2−d as |x|→∞, for d≥5 for self-avoiding walk, for d≥19 for percolation, and for sufficiently large d for lattice trees and animals. These results are complementary to those of [Ann. Probab. 31 (2003) 349–408], where spread-out models were considered. In the course of the proof, we also provide a sufficient (and rather sharp if d>4) condition under which the two-point function of a random walk on ℤd is asymptotic to const.|x|2−d as |x|→∞.

Article information

Source
Ann. Probab. Volume 36, Number 2 (2008), 530-593.

Dates
First available: 29 February 2008

Permanent link to this document
http://projecteuclid.org/euclid.aop/1204306960

Digital Object Identifier
doi:10.1214/009117907000000231

Mathematical Reviews number (MathSciNet)
MR2393990

Zentralblatt MATH identifier
1142.82006

Subjects
Primary: 82B27: Critical phenomena 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41] 82B43: Percolation [See also 60K35] 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Critical behavior two-point function self-avoiding walk percolation lattice trees and animals lace expansion

Citation

Hara, Takashi. Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. The Annals of Probability 36 (2008), no. 2, 530--593. doi:10.1214/009117907000000231. http://projecteuclid.org/euclid.aop/1204306960.


Export citation

References

  • Aizenman, M. (1997). On the number of incipient spanning clusters. Nuclear. Phys. B 485 551–582.
  • Aizenman, M. and Barsky, D. J. (1987). Sharpness of the phase transition in percolation models. Comm. Math. Phys. 108 489–526.
  • Brydges, D. C. and Spencer, T. (1985). Self-avoiding walk in 5 or more dimensions. Comm. Math. Phys. 97 125–148.
  • 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. Paper in preparation.
  • Hara, T., van der Hofstad, R. and Slade, G. (2003). Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models. Ann. Prob. 31 349–408.
  • Hara, T. and Slade, G. (1990). Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128 333–391.
  • Hara, T. and Slade, G. (1990). On the upper critical dimension of lattice trees and lattice animals. J. Statist. Phys. 59 1469–1510.
  • Hara, T. and Slade, G. (1992). The lace expansion for self-avoiding walk in five or more dimensions. Rev. Math. Phys. 4 235–327.
  • Hara, T. and Slade, G. (1992). Self-avoiding walk in five or more dimensions. I. The critical behaviour. Commun. Math. Phys. 147 101–136.
  • 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.
  • Hughes, B. D. (1995). Random Walks and Random Environments. 1. Random Walks. Oxford Univ. Press.
  • Klarner, D. A. (1967). Cell growth problems. Canad. J. Math. 19 851–863.
  • Klein, D. J. (1981). Rigorous results for branched polymer models with excluded volume. J. Chem. Phys. 75 5186–5189.
  • Lawler, G. F. (1991). Intersections of Random Walks. Birkhäuser, Boston.
  • Lawler, G. F. (1994). A note on Green’s function for random walk in four dimensions. Preprint 94-03, Duke Univ.
  • Lawler, G. F. (2004). Private communication.
  • 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.
  • Reisz, T. (1988). A convergence theorem for lattice Feynman integrals with massless propagators. Comm. Math. Phys. 116 573–606.
  • Reisz, T. (1988). A power counting theorem for Feynman integrals on the lattice. Comm. Math. Phys. 116 81–126.
  • Sakai, A. (2007). Lace expansion for the Ising model. Comm. Math. Phys. 272 283–344.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, New York.
  • Slade, G. (1987). The diffusion of self-avoiding random walk in high dimensions. Comm. Math. Phys. 110 661–683.
  • Slade, G. (2006). The Lace Expansion and Its Applications. Lecture Notes in Math. 1879. Springer, Berlin.
  • Sokal, A. D. (1982). An alternate constructive approach to the ϕ43 quantum field theory, and a possible destructive approach to ϕ44. Ann. Inst. H. Poincaré Sect. A (N.S.) 37 317–398.
  • Uchiyama, K. (1998). Greens’ function for random walks on ZN. Proc. London Math. Soc. 77 215–240.