Mesoscopic description of random walks on combs

TL;DR

This paper investigates how random walks on comb-like structures exhibit different diffusion behaviors, including normal, anomalous, and localized diffusion, by analyzing the effects of branch bias and waiting times.

Contribution

It introduces a generic method to derive transport properties of random walks on combs, accounting for branch bias and renormalizing waiting time distributions.

Findings

Normal diffusion observed under certain conditions

Anomalous diffusion occurs with specific branch characteristics

Diffusion failure or localization identified in some regimes

Abstract

Combs are a simple caricature of various types of natural branched structures, which belong to the category of loopless graphs and consist of a backbone and branches. We study continuous time random walks on combs and present a generic method to obtain their transport properties. The random walk along the branches may be biased, and we account for the effect of the branches by renormalizing the waiting time probability distribution function for the motion along the backbone. We analyze the overall diffusion properties along the backbone and find normal diffusion, anomalous diffusion, and stochastic localization (diffusion failure), respectively, depending on the characteristics of the continuous time random walk along the branches.