This work presents an approach to evaluate the exact value of the fractal dimension of the cutting path $d_f^{CP}$ on hierarchical structures with finite order of ramification. Our approach is based on a renormalization group treatment of the universality class of watersheds. By making use of the self-similar property, we show that $d_f^{CP}$ 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 $d_f^{CP}$ is derived based on a computer algorithm that identifies the length of all possible CP's of the first generation.

• Received 5 March 2013

DOI:https://doi.org/10.1103/PhysRevE.87.042113