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

TL;DR

This paper introduces an exact method to evaluate the fractal dimension of the cutting path in hierarchical networks, using renormalization group techniques and self-similarity properties, specifically applied to watersheds and Wheastone lattices.

Contribution

It provides the first renormalization group analysis of watershed universality class and derives exact cutting path dimensions for hierarchical structures.

Findings

Exact fractal dimension of the cutting path on hierarchical networks

First renormalization group treatment of watersheds

Mathematical proof for Wheastone hierarchical lattice

Abstract

This work presents an approach to evaluate the exact value of the fractal dimension of the cutting path $d^{CP}_f$ 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 $d^{CP}_f$ 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 $d^{CP}_f$ is derived based on an computer algorithm that identifies the length of all possible CP's of the first generation.