### 抄録

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

元の言語 | 英語 |
---|---|

ページ（範囲） | 349-408 |

ページ数 | 60 |

ジャーナル | Annals of Probability |

巻 | 31 |

発行部数 | 1 |

DOI | |

出版物ステータス | 出版済み - 1 1 2003 |

外部発表 | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

### これを引用

*Annals of Probability*,

*31*(1), 349-408. https://doi.org/10.1214/aop/1046294314

**Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models.** / Hara, Takashi; Van der Hofstad, Remco; Slade, Gordon.

研究成果: ジャーナルへの寄稿 › 記事

*Annals of Probability*, 巻. 31, 番号 1, pp. 349-408. https://doi.org/10.1214/aop/1046294314

}

TY - JOUR

T1 - Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models

AU - Hara, Takashi

AU - Van der Hofstad, Remco

AU - Slade, Gordon

PY - 2003/1/1

Y1 - 2003/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037253416&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0037253416&partnerID=8YFLogxK

U2 - 10.1214/aop/1046294314

DO - 10.1214/aop/1046294314

M3 - Article

AN - SCOPUS:0037253416

VL - 31

SP - 349

EP - 408

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 1

ER -