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January 2003 Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
Takashi Hara, Gordon Slade, Remco van der Hofstad
Ann. Probab. 31(1): 349-408 (January 2003). DOI: 10.1214/aop/1046294314

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.

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Takashi Hara. Gordon Slade. Remco van der Hofstad. "Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models." Ann. Probab. 31 (1) 349 - 408, January 2003. https://doi.org/10.1214/aop/1046294314

Information

Published: January 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1044.82006
MathSciNet: MR1959796
Digital Object Identifier: 10.1214/aop/1046294314

Subjects:
Primary: 60K35 , 82B41 , 82B43

Keywords: Critical exponent , Lace expansion , lattice animal , lattice tree , percolation , Self-avoiding walk

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 1 • January 2003
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