On the upper critical dimension of Bernoulli percolation
type: Item
instance of: scholarly article
title: On the upper critical dimension of Bernoulli percolation
zbMATH Open document ID: 0627.60100
DOI: 10.1007/BF01221395
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: We derive a set of inequalities for the d-dimensional independent percolation problem. Assuming the existence of critical exponents, these inequalities imply: \[ f+\nu \geq 1+\beta_ Q,\quad \mu +\nu \geq 1+\beta_ Q,\quad \zeta \geq \min \{1,\nu '/\nu \}, \] where the above exponents are f: the flow constant exponent, \(\nu\) (\(\nu\) '): the correlation length exponent below (above) threshold, \(\mu\) : the surface tension exponent, \(\beta_ Q:\) the backbone density exponent and \(\zeta\) : the chemical distance exponent. Note that all of these inequalities are mean-field bounds, and that they relate the exponent \(\nu\) defined from below the percolation threshold to exponents defined from above threshold. Furthermore, we combine the strategy of the proofs of these inequalities with notions of finite-size scaling to derive: \[ \max \{d\nu,d\nu '\}\geq 1+\beta_ Q, \] where d is the lattice dimension. Since \(\beta_ Q\geq 2\beta\), where \(\beta\) is the percolation density exponent, the final bound implies that, below six dimensions, the standard order parameter and correlation length exponents cannot simultaneously assume their mean- field values; hence an implicit bound on the upper critical dimension: \(d_ c\geq 6\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B43
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4020168
zbMATH Keywords: inequalities
zbMATH Keywords: percolation
zbMATH Keywords: backbone density exponent
zbMATH Keywords: critical dimension
author: Jennifer T. Chayes
author: Lincoln Chayes
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01221395
OpenAlex ID: W2018303271
description: scientific article; zbMATH DE number 4020168