Finite-Dimensional Percolation

In this chapter we apply the scaling theories developed in the one-dimensional system and the infinite-dimensional system to systems of finite dimensions. The lowest dimension with an interesting behavior is two dimensions. Here, we introduce effective ways to measure the cluster number density \(n(s,p)\) in two dimension. We develop the scaling theory for \(n(s,p)\) and demonstrate how to use data-collapse plots as an efficient method to measure the critical exponents. We also demonstrate how we can use the scaling theory for \(n(s,p)\) to derive expressions for the density of the spanning cluster, P, and the average cluster size, S. Finally, we demonstrate how the scaling theory provides scaling relations, that is, relations between exponents, and bounds for the values of the critical exponents.