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

Takashi Hara, Remco Van der Hofstad, Gordon Slade

Research output: Contribution to journalArticle

57 Citations (Scopus)

Abstract

We consider spread-out models of self-avoiding walk, bond percolation, lattice trees and bond lattice animals on ℤd, 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 ∈ ℤd, 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 → ∞. 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.

Original languageEnglish
Pages (from-to)349-408
Number of pages60
JournalAnnals of Probability
Volume31
Issue number1
DOIs
Publication statusPublished - Jan 1 2003
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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