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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.08452 |
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| _version_ | 1866911147360256000 |
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| author | Fourn, Samuel Le Liu, Mike Martineau, Sébastien |
| author_facet | Fourn, Samuel Le Liu, Mike Martineau, Sébastien |
| contents | Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there? |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08452 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Percolative properties of the random coprime colouring Fourn, Samuel Le Liu, Mike Martineau, Sébastien Probability Number Theory Given $u$ and $v$ in $\mathbb{Z}^d$, say that $u$ is visible from $v$ if the segment from $u$ to $v$ contains exactly two elements, which are $u$ and $v$. Take $X$ "uniformly at random in $\mathbb{Z}^d$" and colour each vertex $u$ of $\mathbb{Z}^d$ in white if $u$ is visible from $X$ and in black otherwise. Previous independent works of Pleasants-Huck and of the third author give a precise meaning to this definition. This paper is dedicated to the study of this random colouring from the point of view of percolation theory: given a reasonable graph structure on $\mathbb{Z}^d$, how many infinite black (resp. white) connected components are there? |
| title | Percolative properties of the random coprime colouring |
| topic | Probability Number Theory |
| url | https://arxiv.org/abs/2509.08452 |