Estimates on the effective resistance in a long-range percolation on Zd

We give several estimates on volumes and effective resistances in a long-range percolation on a vertex set of a d-dimensional square lattice. When d = 1, our results imply some kind of discontinuity in the long-range percolation model; more precisely, in the order of the effective resistance. Our another consequence is that, when d >= 2 and s is an element of (d, d + 2), where s is the parameter determining the magnitude of the range, the order of the effective resistance corresponds to the a-stable process with alpha = s-d.