Explosive connectivity and mechanical rigidity in cubic lattice structures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909884223586304 |
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| author | Lau, Trenton Choi, Gary P. T. |
| author_facet | Lau, Trenton Choi, Gary P. T. |
| contents | 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 the development of a new theoretical framework. For connectivity, we rigorously establish the presence of sublinear-width merger-cascade windows for $k\ge 2$, which drive macroscopic jumps in the order parameter and imply a first-order transition. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ monotonically 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 this function becomes non-increasing, thus explaining the observed monotonic efficiency, under two physically-motivated assumptions. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01537 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Explosive connectivity and mechanical rigidity in cubic lattice structures Lau, Trenton Choi, Gary P. T. Statistical Mechanics Materials Science Probability 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 the development of a new theoretical framework. For connectivity, we rigorously establish the presence of sublinear-width merger-cascade windows for $k\ge 2$, which drive macroscopic jumps in the order parameter and imply a first-order transition. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ monotonically 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 this function becomes non-increasing, thus explaining the observed monotonic efficiency, under two physically-motivated assumptions. 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. |
| title | Explosive connectivity and mechanical rigidity in cubic lattice structures |
| topic | Statistical Mechanics Materials Science Probability |
| url | https://arxiv.org/abs/2511.01537 |