Estimates on the effective resistance in a long-range percolation on ${\mathbb{Z}}^d$

Abstract

We give several estimates on \emph{volumes} and \emph{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 $d \ge 2$ and $s \in (d, d+2)$, where $s$ is the parameter determining the magnitude of the range, the order of the effective resistance corresponds to the $\alpha$-stable process with $\alpha =s - d$.