The Annals of Probability
- Ann. Probab.
- Volume 31, Number 1 (2003), 349-408.
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
Abstract
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
Source
Ann. Probab., Volume 31, Number 1 (2003), 349-408.
Dates
First available in Project Euclid: 26 February 2003
Permanent link to this document
https://projecteuclid.org/euclid.aop/1046294314
Digital Object Identifier
doi:10.1214/aop/1046294314
Mathematical Reviews number (MathSciNet)
MR1959796
Zentralblatt MATH identifier
1044.82006
Subjects
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]
Keywords
Critical exponent lace expansion lattice tree lattice animal percolation self-avoiding walk
Citation
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. https://projecteuclid.org/euclid.aop/1046294314
