A fast algorithm for 2D Rigidity Percolation

TL;DR

This paper introduces a highly efficient algorithm for 2D Rigidity Percolation that enables large-scale simulations, precise critical point estimation, and new theoretical insights into cluster merging mechanisms.

Contribution

A novel, fast algorithm combining existing methods and new rigidity theory results for exact identification of rigid clusters in 2D percolation.

Findings

Able to simulate systems with over 500 million nodes

Precisely estimated critical exponents and threshold

Provided new theoretical insights into cluster merging mechanisms

Abstract

Rigidity Percolation is a crucial framework for describing rigidity transitions in amorphous systems. We present a new, efficient algorithm to study central-force Rigidity Percolation in two dimensions. This algorithm combines the Pebble Game algorithm, the Newman-Ziff approach to Connectivity Percolation, as well as novel rigorous results in rigidity theory, to exactly identify rigid clusters over the full bond concentration range, in a time that scales as $N^{1.02}$ for a system of $N$ nodes. We perform extensive numerical simulations with systems larger than $500$ million nodes, far beyond the previous limitations. We obtain new, precise estimates for the critical exponents, $\nu=1.1694(8)$ and $D_f=1.8423(7)$, and locate the critical threshold at $p_c = 0.6602741(4)$. Besides opening the way to further accurate numerical studies of Rigidity Percolation, our work provides new rigorous theoretical insights on specific cluster merging mechanisms that distinguish it from the standard Connectivity Percolation problem.