The Annals of Applied Probability

Existence and uniqueness of infinite components in generic rigidity percolation

Alexander E. Holroyd

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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.

Article information

Source
Ann. Appl. Probab., Volume 8, Number 3 (1998), 944-973.

Dates
First available in Project Euclid: 9 August 2002

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1028903458

Digital Object Identifier
doi:10.1214/aoap/1028903458

Mathematical Reviews number (MathSciNet)
MR1627815

Zentralblatt MATH identifier
0932.60093

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]

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

Citation

Holroyd, Alexander E. Existence and uniqueness of infinite components in generic rigidity percolation. Ann. Appl. Probab. 8 (1998), no. 3, 944--973. doi:10.1214/aoap/1028903458. https://projecteuclid.org/euclid.aoap/1028903458


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