Existence and Uniqueness of Infinite Components in Generic Rigidity Percolation

We consider a percolation configuration on a general lattice in which edges are included independently with probability p. We study the rigidity properties of the resulting configuration, in the sense of generic rigidity in d dimensions. We give a mathematically rigorous treatment of the problem, starting with a definition of an infinite rigid component. We prove that for a broad class of lattices, there exists an infinite rigid component for some p strictly below unity. For the particular case of two-dimensional rigidity on the two-dimensional triangular lattice, we prove firstly that the critical probability for rigidity percolation lies strictly above that for connectivity percolation, and secondly that the infinite rigid component (when it exists) is unique for all but countably many values of p. We conjecture that this uniqueness in fact holds for all p. Some of our arguments could be applied to two-dimensional lattices in more generality.