First-order rigidity transition on Bethe Lattices

TL;DR

This paper introduces and solves tree models for rigidity percolation on Bethe lattices, revealing a first-order transition with critical fluctuations, and highlights discrepancies with traditional mean field theories.

Contribution

It presents a novel tree-based model for rigidity percolation, showing a first-order transition with critical behavior, and compares it to existing mean field approaches.

Findings

Rigidity transition occurs at a critical probability p_c.

The order parameter components are singular at p_c.

Infinite-cluster probability P_infinity shows mixed first-order and critical behavior.

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.