Self-Avoiding Walk on Fractal Complex Networks: Exactly Solvable Cases

TL;DR

This paper analytically investigates the self-avoiding walk on fractal networks, deriving critical exponents and connective constants, and provides an exact solution supporting the idea that universality classes depend on more than just fractal dimension.

Contribution

It offers the first analytical calculation of critical exponents for self-avoiding walks on fractal networks and presents an exact solution for a specific network, challenging existing universality assumptions.

Findings

Critical exponent {} equals the displacement exponent.

Derived the connective constant for the (u,v)-flower.

Supported the conjecture that universality class depends on more than fractal dimension.

Abstract

We study the self-avoiding walk on complex fractal networks called the (u,v)-flower by mapping it to the N-vector model in a generating function formalism. First, we analytically calculate the critical exponent {\nu} and the connective constant by a renormalization-group analysis in arbitrary fractal dimensions. We find that the exponent {\nu} is equal to the displacement exponent, which describes the speed of diffusion in terms of the shortest distance. Second, by obtaining an exact solution for the (u,u)-flower, we provide an example which supports the conjecture that the universality class of the self-avoiding walk on graphs is not determined only by the fractal dimension.