Estimates on the effective resistance in anisotropic long-range percolation on Z2

摘要

We consider a long-range percolation on a vertex set of a two-dimensional square lattice, where the probabilities that horizontally long bonds exist may be different from the probabilities that vertically long bonds exist. We give some estimates on the effective resistance in such a model. The precise order of the effective resistance can be represented by using s, and r, which are the parameters determining the magnitude of the range in the horizontal and vertical directions.