Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909866039181312 |
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| author | Bäumler, Johannes |
| author_facet | Bäumler, Johannes |
| contents | We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $u$ and $v$ are connected with probability asymptotic to $\fracβ{\|u-v\|^{2d}}$ for $\|u-v\|_\infty\geq 2$ and with probability 1 for $\|u-v\|_\infty=1$, where $β\geq 0$ is a parameter. It is proven in [5] that there exists an exponent $θ=θ(d,β) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $x \in \mathbb{Z}^d$ scales like $\|x\|^θ$. We prove that this exponent $θ(d,β)$ is continuous and strictly decreasing as a function in $β$. Furthermore, we show that $θ(d,β)=1-β+o(β)$ for small $β$ in dimension $d=1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_04793 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation Bäumler, Johannes Probability 05C12, 60K35, 82B27, 82B43 We study independent long-range percolation on $\mathbb{Z}^d$ where the vertices $u$ and $v$ are connected with probability asymptotic to $\fracβ{\|u-v\|^{2d}}$ for $\|u-v\|_\infty\geq 2$ and with probability 1 for $\|u-v\|_\infty=1$, where $β\geq 0$ is a parameter. It is proven in [5] that there exists an exponent $θ=θ(d,β) \in \left(0,1\right]$ such that the graph distance between the origin $\mathbf{0}$ and $x \in \mathbb{Z}^d$ scales like $\|x\|^θ$. We prove that this exponent $θ(d,β)$ is continuous and strictly decreasing as a function in $β$. Furthermore, we show that $θ(d,β)=1-β+o(β)$ for small $β$ in dimension $d=1$. |
| title | Behavior of the distance exponent for $\frac{1}{|x-y|^{2d}}$ long-range percolation |
| topic | Probability 05C12, 60K35, 82B27, 82B43 |
| url | https://arxiv.org/abs/2208.04793 |