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 , where an edge connects and independently with probability 1 for and with probability for some fixed . 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 and from the interval to (conditioned on no edge joining and ) both have a polynomial lower bound in . Our bound holds for all and thus rules out a potential phase transition (around ) which seemed to be a reasonable possibility.