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

Takashi Hara, Gordon Slade

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1244-1293
Number of pages50
JournalJournal of Mathematical Physics
Volume41
Issue number3
DOIs
Publication statusPublished - Jan 1 2000
Externally publishedYes

Fingerprint

Brownian Excursion
Scaling Limit
High-dimensional
scaling
complex variables
Complex Variables
Error term
Critical Exponents
Critical point
Nearest Neighbor
critical point
Magnetic Field
exponents
Converge
Imply
expansion
magnetic fields

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.

In: Journal of Mathematical Physics, Vol. 41, No. 3, 01.01.2000, p. 1244-1293.

Research output: Contribution to journalArticle

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