### Abstract

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.

Original language | English |
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Pages (from-to) | 337-385 |

Number of pages | 49 |

Journal | Probability Theory and Related Fields |

Volume | 86 |

Issue number | 3 |

DOIs | |

Publication status | Published - Sep 1 1990 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

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

Research output: Contribution to journal › Article

*Probability Theory and Related Fields*, vol. 86, no. 3, pp. 337-385. https://doi.org/10.1007/BF01208256

}

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.

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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 -