First-order rigidity transition on Bethe Lattices

Abstract

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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References in corpus (2)

• Stressed backbone and elasticity of random central-force systems
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