Representation of Archimedean Networks and Inclusion: Computational Applications to Percolation and Network Transitions
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| Format: | Preprint |
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2025
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| author | Torres, Auro Anibal Ramirez-Pastor, José Antonio |
| author_facet | Torres, Auro Anibal Ramirez-Pastor, José Antonio |
| contents | We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling enables efficient and systematic implementation of computational models on these lattices, without relying on ad hoc indexing, and provides a versatile framework for future studies on regular tilings. As an application, we obtain, for each Archimedean lattice, the phase diagrams generated by monomer deposition for the site-bond percolation models known in the literature as $S \cup B$ and $S \cap B$. We show that these diagrams are ordered in phase space according to the partial and total inclusion relations among the lattices, as demonstrated by Parviainen et al. (2003). Furthermore, for the lattice pairs (3.6.3.6)/(3.4.6.4) and $(3^3.4^2)/(3^2.4.3.4)$, we observe an inversion between the pure site and bond percolation thresholds. We show that this phenomenon is linked to the crossing of their respective phase diagrams, highlighting the sensitivity of critical behavior to lattice topology. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_12229 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation of Archimedean Networks and Inclusion: Computational Applications to Percolation and Network Transitions Torres, Auro Anibal Ramirez-Pastor, José Antonio Statistical Mechanics 82B43, 60K35, 82C20 We present an alternative geometric representation for the eleven Archimedean lattices, in which each site and bond is uniquely labeled by an ordered pair of integers and characterized via a modular function. This structured labeling enables efficient and systematic implementation of computational models on these lattices, without relying on ad hoc indexing, and provides a versatile framework for future studies on regular tilings. As an application, we obtain, for each Archimedean lattice, the phase diagrams generated by monomer deposition for the site-bond percolation models known in the literature as $S \cup B$ and $S \cap B$. We show that these diagrams are ordered in phase space according to the partial and total inclusion relations among the lattices, as demonstrated by Parviainen et al. (2003). Furthermore, for the lattice pairs (3.6.3.6)/(3.4.6.4) and $(3^3.4^2)/(3^2.4.3.4)$, we observe an inversion between the pure site and bond percolation thresholds. We show that this phenomenon is linked to the crossing of their respective phase diagrams, highlighting the sensitivity of critical behavior to lattice topology. |
| title | Representation of Archimedean Networks and Inclusion: Computational Applications to Percolation and Network Transitions |
| topic | Statistical Mechanics 82B43, 60K35, 82C20 |
| url | https://arxiv.org/abs/2507.12229 |