Exact evaluation of the cutting path length in a percolation model on a hierarchical network

Abstract: This work presents an approach to evaluate the exact value of the fractal dimension of the cutting path $d<sup>{CP}_f$ on hierarchical structures with finite order of ramification. This represents the first renormalization group treatment of the universality class of watersheds. By making use of the self-similar property, we show that $d<sup>{CP}_f$ depends only on the average cutting path (CP) of the first generation of the structure. For the simplest Wheastone hierarchical lattice (WHL), we present a mathematical proof. For a larger WHL structure, the exact value of $d<sup>{CP}_f$ is derived based on an computer algorithm that identifies the length of all possible CP's of the first generation.