Rigidity percolation and boundary conditions.
type: Item
instance of: scholarly article
title: Rigidity percolation and boundary conditions.
zbMATH Open document ID: 1062.60098
author: A. E. Holroyd
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: The author discusses rigidity properties of components in the bond percolation on the two-dimensional triangular lattice \textbf{T}. While finite rigid components here are straightforward to define, the notion of infinite rigid components is not so obvious. In this paper, the latter are defined as limits of rigid components in a finite region of the lattice with specified boundary conditions (i.e., with fixed states of edges outside the region) as the region approaches \textbf{T}. Among various boundary conditions there are two ``extremal'' choices -- the ``free'' and ``wired'' boundary conditions -- with all edges outside the region being closed and open, respectively. While theoretically such choices might give rise to different definitions of rigidity, the main result of the paper claims that, in two dimensions, the corresponding critical probabilities coincide. Moreover, it is shown that for all values of the tuning parameter \(p\) (with possible exception of the common critical value) the sets of ``free'' and ``wired'' rigid components are identical, almost surely w.r.t.\ the percolation measure \({\mathbf P}_p\).
reviewed by: Ostap Hryniv
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 52C25
Mathematics Subject Classification ID: 82B43
zbMATH DE Number: 1906205
zbMATH Keywords: percolation
zbMATH Keywords: rigidity
zbMATH Keywords: boundary conditions.
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345395
OpenAlex ID: W1986017325
cites work: Q4279195
cites work: Percolation
cites work: Entanglement in percolation
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cites work: Existence of a phase transition for entanglement percolation
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cites work: Q4111952
DOI: 10.1214/AOAP/1015345395
description: scientific article; zbMATH DE number 1906205