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November 2001 Rigidity Percolation and Boundary Conditions
Alexander E. Holroyd
Ann. Appl. Probab. 11(4): 1063-1078 (November 2001). DOI: 10.1214/aoap/1015345395


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 $\mathrm{r}^0$-clusters and $\mathrm{r}^1$-clusters, which correspond to free boundary conditions and wired boundary conditions respectively. The definition of an $\mathrm{r}^ 0$-cluster turns out to be equivalent to the definition of a rigid component used in earlier papers by Holroyd and Häggström. We define two critical probabilities, associated with the appearance of infinite $\mathrm{r}^0$-clusters and infinite $\mathrm{r}^1$-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 $\mathrm{r}^ 0$-clusters equals the set of $\mathrm{r}^ 1$-clusters almost surely. It is an open problem to determine what happens at the critical probability.


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Alexander E. Holroyd. "Rigidity Percolation and Boundary Conditions." Ann. Appl. Probab. 11 (4) 1063 - 1078, November 2001.


Published: November 2001
First available in Project Euclid: 5 March 2002

zbMATH: 1062.60098
MathSciNet: MR1878290
Digital Object Identifier: 10.1214/aoap/1015345395

Primary: 60K35
Secondary: 05B35 , 52C25 , 82B43

Keywords: boundary conditions , percolation , rigidity , Rigidity percolation

Rights: Copyright © 2001 Institute of Mathematical Statistics


Vol.11 • No. 4 • November 2001
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