Triangle Percolation on the Grid
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arXiv
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| Main Authors: | , , , , , , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914328567873536 |
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| author | Araujo, Igor Frederickson, Bryce Krueger, Robert A. Lidický, Bernard McAllister, Tyrrell B. Pfender, Florian Spiro, Sam Stucky, Eric Nathan |
| author_facet | Araujo, Igor Frederickson, Bryce Krueger, Robert A. Lidický, Bernard McAllister, Tyrrell B. Pfender, Florian Spiro, Sam Stucky, Eric Nathan |
| contents | We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subseteq \mathbb{R}^2$ with its vertices in $\mathbb{Z}^2$ such that $T$ contains exactly four points of $\mathbb{Z}^2$ and exactly three points of $X$. In this case, we add the missing lattice point of $T$ to $X$, and we repeat until no such triangle exists. We study the limit sets $S$, the sets stable under this process, including determining their possible densities and some of their structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_15402 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Triangle Percolation on the Grid Araujo, Igor Frederickson, Bryce Krueger, Robert A. Lidický, Bernard McAllister, Tyrrell B. Pfender, Florian Spiro, Sam Stucky, Eric Nathan Combinatorics We consider a geometric percolation process partially motivated by recent work of Hejda and Kala. Specifically, we start with an initial set $X \subseteq \mathbb{Z}^2$, and then iteratively check whether there exists a triangle $T \subseteq \mathbb{R}^2$ with its vertices in $\mathbb{Z}^2$ such that $T$ contains exactly four points of $\mathbb{Z}^2$ and exactly three points of $X$. In this case, we add the missing lattice point of $T$ to $X$, and we repeat until no such triangle exists. We study the limit sets $S$, the sets stable under this process, including determining their possible densities and some of their structure. |
| title | Triangle Percolation on the Grid |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2303.15402 |