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

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dc.creatorHara, Takashi
dc.creatorvan der Hofstad, Remco
dc.creatorSlade, Gordon
dc.date2000-11-28
dc.date.accessioned2026-07-07T04:28:09Z
dc.date.available2026-07-07T04:28:09Z
dc.descriptionWe consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on the d-dimensional hyper cubic lattice 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, 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 goes to infinity. 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.
dc.description53 pages, 9 figures, in AMS-LaTeX
dc.identifierhttps://arxiv.org/abs/math-ph/0011046
dc.identifierhttp://arxiv.org/abs/math-ph/0011046
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/56680
dc.subjectMathematical Physics
dc.subject82B43; 60K35
dc.titleCritical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models
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