Activated random walk on the comb

TL;DR

This paper extends the layer percolation method to activated random walk on the comb graph, providing bounds on critical density and evidence of different critical densities on the spine and teeth.

Contribution

It develops an analog of layer percolation for the comb graph and uses it to analyze critical densities, advancing understanding of activated random walk on complex graphs.

Findings

Bounds on critical density for the comb graph.

Evidence of different critical densities on spine and teeth.

Critical densities are smaller than on the interval.

Abstract

The density conjecture for activated random walk on the interval was recently resolved using a new tool called layer percolation. As a step towards understanding how layer percolation extends to activated random walk on more complex graphs, we develop its analog for the comb graph and use it to prove bounds on the critical density. Additionally, we provide simulation evidence suggesting that the comb has different critical densities on its spine and teeth, both of which are smaller than the critical density for the interval.