Continuity of the critical value and a shape theorem for long-range percolation
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866917443548479488 |
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| author | Bäumler, Johannes |
| author_facet | Bäumler, Johannes |
| contents | Consider supercritical long-range percolation on $\Z^d$ where two vertices $x,y \in \Z^d$ are connected with probability asymptotic to $\|x-y\|^{-s}$ for some $s>2d$. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within $n$ steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\geq 2$, the critical value depends continuously on the precise specifications of the model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_04099 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Continuity of the critical value and a shape theorem for long-range percolation Bäumler, Johannes Probability Mathematical Physics 82B43, 60K35, 05C12, 05C81 Consider supercritical long-range percolation on $\Z^d$ where two vertices $x,y \in \Z^d$ are connected with probability asymptotic to $\|x-y\|^{-s}$ for some $s>2d$. Conditioned that the origin is in the infinite cluster, we prove a shape theorem for the set of points that can be reached within $n$ steps from the origin. As part of the proof, we show that for long-range percolation with polynomially decaying connection probabilities in dimensions $d\geq 2$, the critical value depends continuously on the precise specifications of the model. |
| title | Continuity of the critical value and a shape theorem for long-range percolation |
| topic | Probability Mathematical Physics 82B43, 60K35, 05C12, 05C81 |
| url | https://arxiv.org/abs/2312.04099 |