Decay of connection probability in high-dimensional continuum percolation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916054320545792 |
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| author | Dickson, Matthew Liu, Yucheng |
| author_facet | Dickson, Matthew Liu, Yucheng |
| contents | We study a percolation model on $\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \to \infty$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\mathbb Z^d$ in $d \ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara's induction argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19288 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decay of connection probability in high-dimensional continuum percolation Dickson, Matthew Liu, Yucheng Probability Mathematical Physics 60K35, 82B21, 82B27, 82B43 We study a percolation model on $\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \to \infty$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\mathbb Z^d$ in $d \ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara's induction argument. |
| title | Decay of connection probability in high-dimensional continuum percolation |
| topic | Probability Mathematical Physics 60K35, 82B21, 82B27, 82B43 |
| url | https://arxiv.org/abs/2507.19288 |