### Abstract

For independent nearest-neighbor bond percolation on ℤ^{d} with d≫6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n^{-3/2-∈} with ∈>0. This is a strong version of the statement that the critical exponent δ is given by δ= 2.

Original language | English |
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Pages (from-to) | 1244-1293 |

Number of pages | 50 |

Journal | Journal of Mathematical Physics |

Volume | 41 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2000 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

**The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion.** / Hara, Takashi; Slade, Gordon.

Research output: Contribution to journal › Article

*Journal of Mathematical Physics*, vol. 41, no. 3, pp. 1244-1293. https://doi.org/10.1063/1.533186

}

TY - JOUR

T1 - The scaling limit of the incipient infinite cluster in high-dimensional percolation. II. Integrated super-Brownian excursion

AU - Hara, Takashi

AU - Slade, Gordon

PY - 2000/1/1

Y1 - 2000/1/1

N2 - For independent nearest-neighbor bond percolation on ℤd with d≫6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n-3/2, plus an error term of order n-3/2-∈ with ∈>0. This is a strong version of the statement that the critical exponent δ is given by δ= 2.

AB - For independent nearest-neighbor bond percolation on ℤd with d≫6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n-3/2, plus an error term of order n-3/2-∈ with ∈>0. This is a strong version of the statement that the critical exponent δ is given by δ= 2.

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U2 - 10.1063/1.533186

DO - 10.1063/1.533186

M3 - Article

AN - SCOPUS:0034345450

VL - 41

SP - 1244

EP - 1293

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 3

ER -