Mean-field critical behaviour for correlation length for percolation in high dimensions
type: Item
instance of: scholarly article
title: Mean-field critical behaviour for correlation length for percolation in high dimensions
zbMATH Open document ID: 0685.60102
DOI: 10.1007/BF01208256
author: Takashi Hara
publication date: 1990-01-01T00:00:00Z
review text: We prove that the correlation length \(\xi\) of independent bond percolation models exhibits mean-field type critical behaviour (i.e. \(\xi (p)\sim (p_ c-p)^{-1/2}\) as \(p\nearrow p_ c)\) in two situations: i) for nearest-neighbour independent bond percolation models on a d- dimensional hypercubic lattice \({\mathbb{Z}}^ d\), 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 which was used previously to prove the triangle condition and hence mean-field behaviour of the critical exponents \(\gamma\),\(\beta\),\(\delta\),\(\Delta\) and \(\nu_ 2\) for the above two cases.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 4123029
zbMATH Keywords: correlation length
zbMATH Keywords: bond percolation models
zbMATH Keywords: nearest-neighbour model
zbMATH Keywords: critical exponents
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01208256
OpenAlex ID: W2022947584
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 4123029