Polynomial lower bound on the effective resistance for the one‐dimensional critical long‐range percolation
In this work, we study the critical long‐range percolation (LRP) on Z$\mathbb {Z}$, where an edge connects i$i$ and j$j$ independently with probability 1 for |i−j|=1$|i-j|=1$ and with probability 1−exp{−β∫ii+1∫jj+1|u−v|−2dudv}$1-\exp \lbrace -\beta \int _i^{i+1}\int _j^{j+1}|u-v|^{-2}{\rm d}u{\rm d}...
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| Vydáno v: | Communications on pure and applied mathematics Ročník 78; číslo 7; s. 1251 - 1284 |
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| Hlavní autoři: | , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
John Wiley and Sons, Limited
01.07.2025
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| Témata: | |
| ISSN: | 0010-3640, 1097-0312 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | In this work, we study the critical long‐range percolation (LRP) on Z$\mathbb {Z}$, where an edge connects i$i$ and j$j$ independently with probability 1 for |i−j|=1$|i-j|=1$ and with probability 1−exp{−β∫ii+1∫jj+1|u−v|−2dudv}$1-\exp \lbrace -\beta \int _i^{i+1}\int _j^{j+1}|u-v|^{-2}{\rm d}u{\rm d}v\rbrace$ for some fixed β>0$\beta >0$. 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$[-N, N]^c$ and from the interval [−N,N]$[-N,N]$ to [−2N,2N]c$[-2N,2N]^c$ (conditioned on no edge joining [−N,N]$[-N,N]$ and [−2N,2N]c$[-2N,2N]^c$) both have a polynomial lower bound in N$N$. Our bound holds for all β>0$\beta >0$ and thus rules out a potential phase transition (around β=1$\beta = 1$) which seemed to be a reasonable possibility. |
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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0010-3640 1097-0312 |
| DOI: | 10.1002/cpa.22243 |