Bi-infinite incipient cluster in high dimensions
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913882978648064 |
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| author | Cabezas, Manuel Fribergh, Alexander Heydenreich, Markus Járai, Antal A. |
| author_facet | Cabezas, Manuel Fribergh, Alexander Heydenreich, Markus Járai, Antal A. |
| contents | We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity. This limiting procedure gives rise to a new percolation measure that locally resembles critical percolation but is concentrated on configurations with two disjoint infinite occupied paths. We coin this the bi-infinite incipient percolation cluster. It is mutually singular with respect to incipient infinite clusters that have been constructed in the literature. We achieve the construction through a double lace expansion of the cluster. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bi-infinite incipient cluster in high dimensions Cabezas, Manuel Fribergh, Alexander Heydenreich, Markus Járai, Antal A. Probability 60K35, 82B43 We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity. This limiting procedure gives rise to a new percolation measure that locally resembles critical percolation but is concentrated on configurations with two disjoint infinite occupied paths. We coin this the bi-infinite incipient percolation cluster. It is mutually singular with respect to incipient infinite clusters that have been constructed in the literature. We achieve the construction through a double lace expansion of the cluster. |
| title | Bi-infinite incipient cluster in high dimensions |
| topic | Probability 60K35, 82B43 |
| url | https://arxiv.org/abs/2506.06559 |