Percolation of words on the hypercubic lattice with one-dimensional long-range interactions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916546153021440 |
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| author | Gomes, Pablo A. Lima, Otávio Silva, Roger W C |
| author_facet | Gomes, Pablo A. Lima, Otávio Silva, Roger W C |
| contents | We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter $0$ or $1$ according to Bernoulli r.v.'s with parameter $p$. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on $\mathbb{Z}^{d-1} \times \mathbb{Z}$, $d\geq 3$, where each edge parallel to $\mathbb{Z}^{d-1}$ has length one and is open with probability $ε$, while edges of length $n$ parallel to $\mathbb{Z}$ are open with probability $p_n$. We prove that if the sum of $p_n$ diverges, then for any $ε$ and $p$, there is a $K$ such that all words are seen from the origin with probability close to $1$, even if all connections with length larger than $K$ are suppressed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_13190 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Percolation of words on the hypercubic lattice with one-dimensional long-range interactions Gomes, Pablo A. Lima, Otávio Silva, Roger W C Probability We investigate the problem of percolation of words in a random environment. To each vertex, we independently assign a letter $0$ or $1$ according to Bernoulli r.v.'s with parameter $p$. The environment is the resulting graph obtained from an independent long-range bond percolation configuration on $\mathbb{Z}^{d-1} \times \mathbb{Z}$, $d\geq 3$, where each edge parallel to $\mathbb{Z}^{d-1}$ has length one and is open with probability $ε$, while edges of length $n$ parallel to $\mathbb{Z}$ are open with probability $p_n$. We prove that if the sum of $p_n$ diverges, then for any $ε$ and $p$, there is a $K$ such that all words are seen from the origin with probability close to $1$, even if all connections with length larger than $K$ are suppressed. |
| title | Percolation of words on the hypercubic lattice with one-dimensional long-range interactions |
| topic | Probability |
| url | https://arxiv.org/abs/2202.13190 |