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 on $\mathbb{Z}$, where an edge connects $i$ and $j$ independently with probability $1-\exp \{-\beta \mathrm{\mid }i-j{\mathrm{\mid }}^{-2}\}$ for some fixed $\beta >0$. 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 $[-N,N{]}^c$ 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 $\beta >0$ and thus rules out a potential phase transition (around $\beta =1$) which seemed to be a reasonable possibility.