Abstract：

In this work, we study the critical long-range percolation on ℤ, where a long-range edge connects 𝑖 and 𝑗 independently with probability 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 resistance from the origin 0 to [-N, N]^c has a polynomial lower bound in N. Our bound holds for any 𝛽 > 0 and thus rules out a potential phase transition (around 𝛽=1) which seemed to be a reasonable possibility. This is a joint work with Jian Ding and Zherui Fan.