You are not currently logged in.
Access JSTOR through your library or other institution:
If You Use a Screen ReaderThis content is available through Read Online (Free) program, which relies on page scans. 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
Alexander E. Holroyd
The Annals of Applied Probability
Vol. 11, No. 4 (Nov., 2001), pp. 1063-1078
Published by: Institute of Mathematical Statistics
Stable URL: http://www.jstor.org/stable/2699909
Page Count: 16
You can always find the topics here!Topics: Vertices, Boundary conditions, Graph theory, Stiffness, Matroids, Degrees of freedom, Mathematics, Connectivity, Mathematical lattices, Integers
Were these topics helpful?See somethings inaccurate? Let us know!
Select the topics that are inaccurate.
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.
Preview not available
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.
The Annals of Applied Probability © 2001 Institute of Mathematical Statistics