Uniqueness in two-dimensional rigidity percolation

For bond percolation on the two-dimensional triangular lattice with arbitra ry retention parameter p is an element of [0, 1], we show that the number o f infinite rigid components is a.s. at most 1. This proves a, conjecture by Holroyd. Further results, concerning simultaneous uniqueness, and continui ty (in p) of the probability that a given edge is in an infinite rigid comp onent, are also obtained.