Renormalization group calculation of distribution functions: Structural properties for percolation clusters

Abstract

Distribution functions of properties of critical percolation clusters are calculated using the ``$H$-cell'' real space renormalization group (RG). We consider structural properties which span two terminals on percolation clusters. These include the lengths of (minimal, average edge-to-edge, longest) self-avoiding walks, the number of the singly connected bonds, and the masses of percolation clusters, as well as of the backbone. We show that the RG corresponds to a (Galton-Watson) branching process, and apply theorems developed in the mathematical literature. We derive recursion relations for the distribution functions, and find exact functional forms for their asymptotic tail behavior at both small and large arguments. The results for the minimal paths have implications on the (multifractal) distribution of wave functions, while the singly connected bonds determine the moments of Ising correlations on these clusters. Our results compare well with existing simulation data.