Rigidity Percolation and Boundary Conditions

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 $\mathrm{r}^0$-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.