Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915536203415552 |
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| author | Bäumler, Johannes |
| author_facet | Bäumler, Johannes |
| contents | Consider independent long-range percolation on $\mathbb{Z}^d$ for $d\geq 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\geq 3$, we show that if the expected degree of the origin is at least $10^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_00303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$ Bäumler, Johannes Probability 82B43, 60K35 Consider independent long-range percolation on $\mathbb{Z}^d$ for $d\geq 3$. Assuming that the expected degree of the origin is infinite, we show that there exists an $N\in \mathbb{N}$ such that an infinite open cluster remains after deleting all edges of length at least $N$. For the isotropic case in dimensions $d\geq 3$, we show that if the expected degree of the origin is at least $10^{400}$, then there exists an infinite open cluster almost surely. We also use these results to prove corresponding statements for the long-range $q$-states Potts model. |
| title | Truncation of long-range percolation with non-summable interactions in dimensions $d\geq 3$ |
| topic | Probability 82B43, 60K35 |
| url | https://arxiv.org/abs/2410.00303 |