The polynomial growth of effective resistances in one-dimensional critical long-range percolation

Abstract

We study the critical long-range percolation on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp \{-\beta {\int }_i^{i+1}{\int }_j^{j+1}|u-v{|}^{-2}\mathrm{d}u\mathrm{d}v\}$ for $|i-j|>1$ for some fixed $\beta >0$ 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 $n^{\delta (\beta )}$ for some $\delta (\beta )\in (0,1)$.

68 pages, 11 figures