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-exp\-β |i-j|⁻²\ for some fixed β>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 β>0 and thus rules out a potential phase transition (around β = 1) which seemed to be a reasonable possibility.