Effect of Loops on the Vibrational Spectrum of Percolation Network

TL;DR

This paper investigates how adding loops to a critical percolation cluster affects its vibrational and elastic properties, revealing a potential new universality class due to loop formation.

Contribution

It introduces a numerical analysis showing that large loops significantly alter spectral and walk dimensions, indicating a new universality class in percolation networks.

Findings

Spectral dimension and walk dimension change abruptly with large loop formation.

Adding smaller loops has minimal effect on these exponents.

Loop formation leads to a new universality class in percolation networks.

Abstract

We study the effects of adding loops to a critical percolation cluster on the diffusional, and equivalently, (scalar) elastic properties of the fractal network. From the numerical calculations of the eigenspectrum of the transition probability matrix, we find that the spectral dimension $d_s$ and the walk dimension $d_w$ change suddenly as soon as the floppy ends of a critical percolation cluster are connected together to form relatively large loops, and that the additional inclusion of successively smaller loops only change these exponents little if at all. This suggests that there is a new universality class associated with the loop-enhanced percolation problem.