### 抄録

Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(p_{c}-p)^{-1/2} as p↗p_{c}) in two situations: i) for nearest-neighbour independent bond percolation models on a d-dimensional hypercubic lattice ℤ^{d}, with d sufficiently large, and ii) for a class of "spread-out" independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν_{2} for the above two cases.

元の言語 | 英語 |
---|---|

ページ（範囲） | 337-385 |

ページ数 | 49 |

ジャーナル | Probability Theory and Related Fields |

巻 | 86 |

発行部数 | 3 |

DOI | |

出版物ステータス | 出版済み - 9 1 1990 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### これを引用

**Mean-field critical behaviour for correlation length for percolation in high dimensions.** / Hara, Takashi.

研究成果: ジャーナルへの寄稿 › 記事

}

TY - JOUR

T1 - Mean-field critical behaviour for correlation length for percolation in high dimensions

AU - Hara, Takashi

PY - 1990/9/1

Y1 - 1990/9/1

N2 - Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(pc-p)-1/2 as p↗pc) in two situations: i) for nearest-neighbour independent bond percolation models on a d-dimensional hypercubic lattice ℤd, with d sufficiently large, and ii) for a class of "spread-out" independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν2 for the above two cases.

AB - Extending the method of [27], we prove that the corrlation length ξ of independent bond percolation models exhibits mean-field type critical behaviour (i.e. ξ(p∼(pc-p)-1/2 as p↗pc) in two situations: i) for nearest-neighbour independent bond percolation models on a d-dimensional hypercubic lattice ℤd, with d sufficiently large, and ii) for a class of "spread-out" independent bond percolation models, which are believed to belong to the same universality class as the nearest-neighbour model, in more than six dimensions. The proof is based on, and extends, a method developed in [27], where it was used to prove the triangle condition and hence mean-field behaviour of the critical exponents γ, β, δ, Δ and ν2 for the above two cases.

UR - http://www.scopus.com/inward/record.url?scp=0001638218&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001638218&partnerID=8YFLogxK

U2 - 10.1007/BF01208256

DO - 10.1007/BF01208256

M3 - Article

AN - SCOPUS:0001638218

VL - 86

SP - 337

EP - 385

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3

ER -