Truncated long-range percolation of words on the square lattice
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866918421945384960 |
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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 study mixed long-range percolation on the square lattice. Each vertical edge of unit length is independently open with probability $\varepsilon$, and each horizontal edge of length $i$ is independently open with probability $p_i$. Also, each vertex is assigned independently a random variable taking values 1 and 0 with probability $p$ and $1-p$, respectively. We prove that for a broad class of anisotropic long-range percolation models for which connection probabilities $p_i$ satisfy some regularity conditions, all words (semi-infinite binary sequences) are seen simultaneously from the origin with positive probability, even if all edges with length larger than some constant (depending on $\varepsilon$, $p$, and on the sequence $(p_i)$) are suppressed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19379 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Truncated long-range percolation of words on the square lattice Gomes, Pablo A. Lima, Otávio Silva, Roger W. C. Probability We study mixed long-range percolation on the square lattice. Each vertical edge of unit length is independently open with probability $\varepsilon$, and each horizontal edge of length $i$ is independently open with probability $p_i$. Also, each vertex is assigned independently a random variable taking values 1 and 0 with probability $p$ and $1-p$, respectively. We prove that for a broad class of anisotropic long-range percolation models for which connection probabilities $p_i$ satisfy some regularity conditions, all words (semi-infinite binary sequences) are seen simultaneously from the origin with positive probability, even if all edges with length larger than some constant (depending on $\varepsilon$, $p$, and on the sequence $(p_i)$) are suppressed. |
| title | Truncated long-range percolation of words on the square lattice |
| topic | Probability |
| url | https://arxiv.org/abs/2412.19379 |