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August 1998 Existence and uniqueness of infinite components in generic rigidity percolation
Alexander E. Holroyd
Ann. Appl. Probab. 8(3): 944-973 (August 1998). DOI: 10.1214/aoap/1028903458

Abstract

We consider a percolation configuration on a general lattice in which edges are included independently with probability p. We study the rigidity properties of the resulting configuration, in the sense of generic rigidity in d dimensions. We give a mathematically rigorous treatment of the problem, starting with a definition of an infinite rigid component. We prove that, for a broad class of lattices, there exists an infinite rigid component for some p strictly below unity. For the particular case of two-dimensional rigidity on the two-dimensional triangular lattice, we prove first that the critical probability for rigidity percolation lies strictly above that for connectivity percolation and second that the infinite rigid component (when it exists) is unique for all but countably many values of p. We conjecture that this uniqueness in fact holds for all p. Some of our arguments could be applied to two-dimensional lattices in more generality.

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Alexander E. Holroyd. "Existence and uniqueness of infinite components in generic rigidity percolation." Ann. Appl. Probab. 8 (3) 944 - 973, August 1998. https://doi.org/10.1214/aoap/1028903458

Information

Published: August 1998
First available in Project Euclid: 9 August 2002

zbMATH: 0932.60093
MathSciNet: MR1627815
Digital Object Identifier: 10.1214/aoap/1028903458

Subjects:
Primary: 60K35
Secondary: 05C10 , 82B43

Keywords: critical points , enhancements , graph rigidity , infinite components , percolation , Rigidity percolation , uniqueness

Rights: Copyright © 1998 Institute of Mathematical Statistics

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Vol.8 • No. 3 • August 1998
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