Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation
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arXiv
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| Format: | Preprint |
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2024
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| author | Ding, Jian Fan, Zherui Huang, Lu-Jing |
| author_facet | Ding, Jian Fan, Zherui Huang, Lu-Jing |
| contents | In this work, we study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β|i-j|^{-2}\}$ for some fixed $β>0$. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to $[-N, N]^c$ and from the interval $[-N,N]$ to $[-2N,2N]^c$ (conditioned on no edge joining $[-N,N]$ and $[-2N,2N]^c$) both have a polynomial lower bound in $N$. Our bound holds for all $β>0$ and thus rules out a potential phase transition (around $β= 1$) which seemed to be a reasonable possibility. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03460 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation Ding, Jian Fan, Zherui Huang, Lu-Jing Probability 60K35, 82B27, 82B43 In this work, we study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β|i-j|^{-2}\}$ for some fixed $β>0$. Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to $[-N, N]^c$ and from the interval $[-N,N]$ to $[-2N,2N]^c$ (conditioned on no edge joining $[-N,N]$ and $[-2N,2N]^c$) both have a polynomial lower bound in $N$. Our bound holds for all $β>0$ and thus rules out a potential phase transition (around $β= 1$) which seemed to be a reasonable possibility. |
| title | Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation |
| topic | Probability 60K35, 82B27, 82B43 |
| url | https://arxiv.org/abs/2405.03460 |