# Biased random walks on combs

## TL;DR

This paper develops analytical methods to study biased random walks on comb structures, calculating spectral dimensions and analyzing transport properties for various random comb ensembles.

## Contribution

It introduces rigorous techniques to exactly determine spectral dimensions and transport behaviors of biased random walks on different random comb models.

## Key findings

- Spectral dimension calculated for various bias scenarios.
- Transport properties along the spine analyzed for specific ensembles.
- Exact results obtained for random combs with infinite teeth and power-law distributions.

## Abstract

We develop rigorous, analytic techniques to study the behaviour of biased random walks on combs. This enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine. Two specific examples of random comb ensembles are discussed; the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths. We also analyze transport properties along the spine for these probability measures.