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Explosive connectivity and mechanical rigidity in cubic lattice structures

Trenton Lau* and Gary P. T. Choi†

  • *Contact author: trentonlau@link.cuhk.edu.hk
  • †Contact author: ptchoi@cuhk.edu.hk
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Phys. Rev. Research 8, 043134 – Published 6 November, 2026

DOI: https://doi.org/10.1103/1fr3-bbxr

Abstract

We study explosive connectivity and mechanical rigidity in three-dimensional cubic lattice structures under Achlioptas-type product-rule dynamics. Our work combines extensive numerical simulation with a theoretical framework based on rigorous finite-size scaling. Using massive-scale simulations up to L=192 (N≈7×106) with 20 000 independent realizations, we demonstrate that for k≥8, the peak susceptibility scales with an exponent of γ=1.000, and the maximum single-step jump stabilizes at a macroscopic fraction. This confirms that while the transition is continuous in the infinite thermodynamic limit, it exhibits the exact finite-size scaling signatures of a first-order discontinuity in finite physical systems. For rigidity, we discover numerically that for richly connected hosts, increasing the number of choices k optimally enhances the efficiency of rigidification. To explain this phenomenon, we propose a theoretical model centered on a conditional progress function that links an edge’s local product-rule score to its global mechanical utility. We show that while local rigidification efficiency monotonically increases, the global rigidity gap exhibits a “Goldilocks” minimum at intermediate k due to the emergence of maximally floppy, treelike components at large k. Altogether, our work provides new insights into the relationship between local dynamics and global connectivity and rigidity in cubic lattice structures via both theory and computation.

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Phys. Rev. Research 8, 043134– Published 6 November, 2026
  • Received 4 November 2025
  • Accepted 19 October 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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