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 ${\mathrm{p}}_{\mathrm{c}}$. The infinite-cluster probability ${\mathrm{P}}_{\mathrm{\infty }}$ is usually first order at ${\mathrm{p}}_{\mathrm{c}}$, except in those cases which are equivalent to connectivity percolation. In many cases, ${\mathrm{P}}_{\mathrm{\infty }}$∼Δ${\mathrm{P}}_{\mathrm{\infty }}$+(p-${\mathrm{p}}_{\mathrm{c}}$ ${)}^{1\mathrm{/}2}$, indicating critical fluctuations superimposed on the first-order jump (Δ${\mathrm{P}}_{\mathrm{\infty }}$). 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.