# Random walks on uniform and non-uniform combs and brushes

## TL;DR

This paper investigates random walks on comb- and brush-like fractal graphs with uniform and non-uniform distributions of side-groups, extending existing models and analyzing sub-diffusive and super-diffusive behaviors through numerical and qualitative methods.

## Contribution

It extends the qualitative method for evaluating sub-diffusion exponents to brushes with fractal bases and analyzes super-diffusive regimes in non-uniform comb-like graphs.

## Key findings

- Extended the sub-diffusion exponent evaluation method to fractal brushes.
- Numerical testing on the Sierpinski brush confirms theoretical predictions.
- Identified super-diffusive transport regimes in non-uniform combs with finite-sized side-groups.

## Abstract

We consider random walks on comb- and brush-like graphs consisting of a base (of fractal dimension $D$) decorated with attached side-groups. The graphs are also characterized by the fractal dimension $D_a$ of a set of anchor points where side-groups are attached to the base. Two types of graphs are considered. Graphs of the first type are uniform in the sense that anchor points are distributed periodically over the base, and thus form a subset of the base with dimension $D_a=D$. Graphs of the second type are decorated with side-groups in a regular yet non-uniform way: the set of anchor points has fractal dimension smaller than that of the base, $D_a<D$. For uniform graphs, a qualitative method for evaluating the sub-diffusion exponent suggested by Forte et al. for combs ($D=1$) is extended for brushes ($D>1$) and numerically tested for the Sierpinski brush (with the base and anchor set built on the same Sierpinski gasket). As an example of nonuniform graphs we consider the Cantor comb composed of a one-dimensional base and side-groups, the latter attached to the former at anchor points forming the Cantor set. A peculiar feature of this and other nonuniform systems is a long-lived regime of super-diffusive transport when side-groups are of a finite size.