Subdiffusive concentration for the chemical distance in Bernoulli percolation
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912553731358720 |
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| author | Can, Van Hao Nguyen, Van Quyet |
| author_facet | Can, Van Hao Nguyen, Van Quyet |
| contents | Considering supercritical Bernoulli percolation on $\mathbb{Z}^d$, Garet and Marchand [GM09] proved a diffusive concentration for the graph distance. In this paper, we sharpen this result by establishing the subdiffusive concentration inequality, which revisits the sublinear bound of the variance proved by Dembin [Dem22] as a consequence. Our approach is inspired by similar work in First-passage percolation [BR08, DHS14], combined with new tools to address the challenge posed by the infinite weight of the model. These tools, including the notion of effective radius and its properties, enable a simple one-step renormalization process as a systematic means of managing the effects of resampling edges. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14834 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subdiffusive concentration for the chemical distance in Bernoulli percolation Can, Van Hao Nguyen, Van Quyet Probability Mathematical Physics Primary 60K37, Secondary 60K35, 82D30, 82B43 Considering supercritical Bernoulli percolation on $\mathbb{Z}^d$, Garet and Marchand [GM09] proved a diffusive concentration for the graph distance. In this paper, we sharpen this result by establishing the subdiffusive concentration inequality, which revisits the sublinear bound of the variance proved by Dembin [Dem22] as a consequence. Our approach is inspired by similar work in First-passage percolation [BR08, DHS14], combined with new tools to address the challenge posed by the infinite weight of the model. These tools, including the notion of effective radius and its properties, enable a simple one-step renormalization process as a systematic means of managing the effects of resampling edges. |
| title | Subdiffusive concentration for the chemical distance in Bernoulli percolation |
| topic | Probability Mathematical Physics Primary 60K37, Secondary 60K35, 82D30, 82B43 |
| url | https://arxiv.org/abs/2406.14834 |