## The Annals of Probability

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

#### 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

https://projecteuclid.org/euclid.aop/1046294314

Digital Object Identifier
doi:10.1214/aop/1046294314

Mathematical Reviews number (MathSciNet)
MR1959796

Zentralblatt MATH identifier
1044.82006

#### 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

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• VANCOUVER, BC CANADA V6T 1Z2 E-MAIL: slade@math.ubc.ca R. VAN DER HOFSTAD STIELTJES INSTITUTE FOR MATHEMATICS DELFT UNIVERSITY OF TECHNOLOGY MEKELWEG 4 2628 CD DELFT THE NETHERLANDS AND DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE TECHNICAL UNIVERSITY EINDHOVEN THE NETHERLANDS E-MAIL: r.w.v.d.hofstsd@tue.nl