First-order rigidity transition on Bethe Lattices

Details First-order rigidity transition on Bethe Lattices First-order rigidity transition on Bethe Lattices

1997-10-13

arxiv.org/abs/cond-mat/9710121v1

Physical Review E 55, 5800 (1997)

Physics

Condensed Matter

Statistical Mechanics

RevTeX, 11 pages + 8 .gif figures

Scientific paper

10.1103/PhysRevE.55.5800

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.

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