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 $\mathbb {Z}$ , where an edge connects i $i$ and j $j$ independently with probability 1 for| i − j | = 1 $|i-j|=1$ and with probability1 − exp { − β ∫ i i + 1∫ j j + 1 | u − v | − 2 d u d v } $1-\exp \lbrace -\beta \int _i^{i+1}\int _j^{j+1}|u-v|^{-2}{\rm d}u{\rm d}v\rbrace$ for some fixedβ > 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[ − 2 N , 2 N ] c $[-2N,2N]^c$ (conditioned on no edge joining[ − N , N ] $[-N,N]$ and[ − 2 N , 2 N ] c $[-2N,2N]^c$ ) both have a polynomial lower bound in N $N$ . Our bound holds for allβ > 0 $\beta >0$ and thus rules out a potential phase transition (aroundβ = 1 $\beta = 1$ ) which seemed to be a reasonable possibility.