Rigidity Percolation and Boundary Conditions

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 Häggström. 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.