An Exact Percolation Point of Surface Filling in a Four-Dimensional Hyper-Cubic Lattice

Abstract

An exact percolation threshold in a four-dimensional hyper-cubic lattice is obtained as p c = 0.5 from geometrical consideration. In a similar way, a macroscopically percolated structure consisting of (2n - n)-dimensional structure is formed at the critical point p c = 0.5 in a 2n-dimensional hyper-cubic lattice, because of the existence of a self-dual lattice in 2n-dimensional space and the orthogonal relation of the (2n - n)-dimensional structures in the original and dual hyper-cubic lattices.