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Rigidity Percolation and Boundary Conditions

Alexander E. Holroyd
The Annals of Applied Probability
Vol. 11, No. 4 (Nov., 2001), pp. 1063-1078
Stable URL: http://www.jstor.org/stable/2699909
Page Count: 16
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Rigidity Percolation and Boundary Conditions
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Abstract

We study the effects of boundary conditions in two-dimensional rigidity percolation. Specifically, we consider generic rigidity in the bond percolation model on the triangular lattice. We introduce a theory of boundary conditions and define two different notions of "rigid clusters," called r0-clusters and r1-clusters, which correspond to free boundary conditions and wired boundary conditions respectively. The definition of an r0-cluster turns out to be equivalent to the definition of a rigid component used in earlier papers by Holroyd and Haggstrom. We define two critical probabilities, associated with the appearance of infinite r0-clusters and infinite r1-clusters respectively, and we prove that these two critical probabilities are in fact equal. Furthermore, we prove that for all parameter values p except possibly this unique critical probability, the set of r0-clusters equals the set of r1-clusters almost surely. It is an open problem to determine what happens at the critical probability.

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