Diffusion in sparse networks: linear to semi-linear crossover

TL;DR

This paper investigates the diffusion properties in sparse networks across different dimensions, challenging previous expectations of a transition to sub-diffusion, and introduces an effective-range-hopping method for analysis.

Contribution

It demonstrates that the diffusion coefficient remains finite in higher dimensions, contrary to prior renormalization-group predictions, and proposes a new effective-range-hopping approach.

Findings

Diffusion coefficient D is finite in higher dimensions.

Contradicts previous renormalization-group predictions.

Introduces an effective-range-hopping method for analysis.

Abstract

We consider random networks whose dynamics is described by a rate equation, with transition rates $w_{nm}$ that form a symmetric matrix. The long time evolution of the system is characterized by a diffusion coefficient $D$. In one dimension it is well known that $D$ can display an abrupt percolation-like transition from diffusion ($D>0$) to sub-diffusion (D=0). A question arises whether such a transition happens in higher dimensions. Numerically $D$ can be evaluated using a resistor network calculation, or optionally it can be deduced from the spectral properties of the system. Contrary to a recent expectation that is based on a renormalization-group analysis, we deduce that $D$ is finite; suggest an "effective-range-hopping" procedure to evaluate it; and contrast the results with the linear estimate. The same approach is useful for the analysis of networks that are described by quasi-one-dimensional sparse banded matrices.