Polynomial lower bound on the effective resistance for the one-dimensional critical long-range percolation

Abstract

In this work, we study the critical long-range percolation (LRP) on $Z$, where an edge connects $i$ and $j$ independently with probability 1 for $|i-j|=1$ and with probability $1-\exp \{-\beta {\int }_i^{i+1}{\int }_j^{j+1}|u-v{|}^{-2}dudv\}$ 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.