FIRST-ORDER RIGIDITY ON CAYLEY-TREES

Tree models for rigidity percolation, in systems with only central for ces, are introduced and solved, A probability vector describes the pro pagation of rigidity outward from a rigid border. All components of th is ''vector order parameter'' are singular at the same rigidity thresh old p(c). The infinite-cluster probability P-infinity is usually first order at p(c), except in those cases which are equivalent to connecti vity percolation. In many cases, P(infinity)similar to Delta P-infinit y+(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 th eories. In an important subclass of tree models ''bootstrap'' percolat ion and rigidity percolation are equivalent.