The polynomial growth of the infinite long-range percolation cluster
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866908528205103104 |
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| author | Bäumler, Johannes |
| author_facet | Bäumler, Johannes |
| contents | We study independent long-range percolation on $\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\|x-y\|>1$ are connected by an edge is proportional to $\fracβ{\|x-y\|^s}$, where $β>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the probability that the origin and $ne_1$ are connected by a path of length $k$ in the critical case $s=2d$. We use these results to determine the Hausdorff dimension of the critical long-range percolation metric that was recently constructed by Ding, Fan, and Huang [14]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14352 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The polynomial growth of the infinite long-range percolation cluster Bäumler, Johannes Probability 05C12, 60K35, 82B27, 82B43 We study independent long-range percolation on $\mathbb{Z}^d$ where the nearest-neighbor edges are always open and the probability that two vertices $x,y$ with $\|x-y\|>1$ are connected by an edge is proportional to $\fracβ{\|x-y\|^s}$, where $β>0$ and $s> 0$ are parameters. We show that the ball of radius $k$ centered at the origin in the graph metric grows polynomially if and only if $s\geq 2d$. For the critical case $s=2d$, we show that the volume growth exponent is inversely proportional to the distance growth exponent. Furthermore, we provide sharp upper and lower bounds on the probability that the origin and $ne_1$ are connected by a path of length $k$ in the critical case $s=2d$. We use these results to determine the Hausdorff dimension of the critical long-range percolation metric that was recently constructed by Ding, Fan, and Huang [14]. |
| title | The polynomial growth of the infinite long-range percolation cluster |
| topic | Probability 05C12, 60K35, 82B27, 82B43 |
| url | https://arxiv.org/abs/2311.14352 |