Infinite-Dimensional Percolation

In this chapter we address the percolation problem on an infinite-dimensional lattice without loops. In this case, it is possible to calculate several of the properties of the percolation system analytically. This allows us to develop a general theory and to develop concepts to be used for finite-dimensional systems. We introduce the infinite-dimensional Bethe lattice for a given coordination number. We find an exact solution for P and the average cluster size S, and use a Taylor-expansion to find an expression for \(n(s,p)\) . The methods and functional forms for \(n(s,p)\) we introduce here, are used to interpret results in finite dimensions.