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  • Open Access

Fast Algorithm for 2D Rigidity Percolation

Nina Javerzat*,‡

Daniele Notarmuzi†,‡

  • Institut für Theoretische Physik, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria

  • *Contact author: nina.javerzat@univ-grenoble-alpes.fr
  • †Contact author: daniele.notarmuzi@tuwien.ac.at
  • ‡These authors contributed equally to this work.
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Phys. Rev. X 16, 031032 – Published 10 August, 2026

DOI: https://doi.org/10.1103/sv88-j2wj

Abstract

Rigidity percolation is a crucial framework for describing rigidity transitions in amorphous systems. We present an 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 N1.02 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 precise estimates for the critical exponents ν=1.1694(8) and Df=1.8423(7) and locate the critical threshold at pc=0.660 274 1(4). Besides opening the way to further accurate numerical studies of rigidity percolation, our work provides rigorous theoretical insights on specific cluster merging mechanisms that distinguish it from the standard connectivity percolation problem.

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Popular Summary

Gels, biological tissues, and granular materials lack obvious structural order, and retain a liquidlike structure even when they behave like solids. Rigidity percolation provides an elegant explanation for the solidification of amorphous media: As more components randomly bind together, a system-spanning, tortuous rigid structure emerges to provide global mechanical stability. However, this process remains poorly understood theoretically and computationally. This work advances both fronts, focusing on two-dimensional networks with purely central forces (e.g., spring networks). First, we determine how rigid structures evolve when a new bond is added to the network. Using these results, we design a fast, near-linear-time algorithm to efficiently simulate the rigidity percolation transition. Finally, we compute high-accuracy critical exponents characterizing behavior near this fluid-to-solid transition. These advances could improve our understanding of network perturbations and aid the design of efficient algorithms to simulate rigidity transitions on networks with noncentral forces, such as frictional grain packings.

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Phys. Rev. X 16, 031032– Published 10 August, 2026
  • Received 2 July 2025
  • Accepted 11 June 2026
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