The polynomial growth of effective resistances in one-dimensional critical long-range percolation
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913880371888128 |
|---|---|
| author | Ding, Jian Fan, Zherui Huang, Lu-Jing |
| author_facet | Ding, Jian Fan, Zherui Huang, Lu-Jing |
| contents | We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 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 grow like $n^{δ(β)}$ for some $δ(β)\in (0,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21378 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The polynomial growth of effective resistances in one-dimensional critical long-range percolation Ding, Jian Fan, Zherui Huang, Lu-Jing Probability 60K35, 82B27, 82B43 We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp\{-β\int_i^{i+1}\int_j^{j+1}|u-v|^{-2}{\rm d} u{\rm d} v\}$ for $|i-j|>1$ for some fixed $β>0$ and with probability 1 for $|i-j|=1$. Viewing this as a random electric network where each edge has a unit conductance, we show that the effective resistances from 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 grow like $n^{δ(β)}$ for some $δ(β)\in (0,1)$. |
| title | The polynomial growth of effective resistances in one-dimensional critical long-range percolation |
| topic | Probability 60K35, 82B27, 82B43 |
| url | https://arxiv.org/abs/2504.21378 |