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 Z, where an edge connects i and j independently with probability 1-\- |i-j|^{-2\} for some fixed >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, Nᶜ and from the interval -N, N to -2N, 2Nᶜ (conditioned on no edge joining -N, N and -2N, 2Nᶜ) both have a polynomial lower bound in N. Our bound holds for all >0 and thus rules out a potential phase transition (around = 1) which seemed to be a reasonable possibility.