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Uniqueness in two-dimensional rigidity percolation

Published online by Cambridge University Press:  06 March 2001

OLLE HÄGGSTRÖM
OLLE HÄGGSTRÖM
Affiliation:
Mathematical Statistics, Chalmers University of Technology, 412 96 Göteborg, Sweden. e-mail: olleh@math.chalmers.sehttp://www.math.chalmers.se/~olleh/
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Abstract

For bond percolation on the two-dimensional triangular lattice with arbitrary retention parameter p ∈ [0, 1], we show that the number of infinite rigid components is a.s. at most 1. This proves a conjecture by Holroyd. Further results, concerning simultaneous uniqueness, and continuity (in p) of the probability that a given edge is in an infinite rigid component, are also obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 1 , January 2001 , pp. 175 - 188
Copyright
2001 Cambridge Philosophical Society

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