First-order rigidity on Cayley trees

Tree models for rigidity percolation, in systems with only central forces, are introduced and solved, A probability vector describes the propagation of rigidity outward from a rigid border. All components of this ''vector order parameter'' are singular at the same rigidity threshold p(c). The infinite-cluster probability P-infinity is usually first order at p(c), except in those cases which are equivalent to connectivity percolation. In many cases, P(infinity)similar to Delta P-infinity+(p-p(c))(1/2), indicating critical fluctuations superimposed on the first-order jump (Delta P-infinity). Our tree models for rigidity are in qualitative disagreement with ''constraint-counting'' mean-field theories. In an important subclass of tree models ''bootstrap'' percolation and rigidity percolation are equivalent.