Tree models for rigidity percolation 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 $P_{ \infty}$ is usually first-order at $p_c$, but often behaves as $P_{ \infty} \sim \Delta P_{ \infty} + (p-p_c)^{ 1/2}$, indicating critical fluctuations superimposed on a first order jump. Our tree models for rigidity are in qualitative disagreement with ``constraint counting'' mean field theories. In an important sub-class of tree models ``Bootstrap'' percolation and rigidity percolation are equivalent.
Categories: cond-mat.stat-mech cond-mat.mtrl-sci