Activated random walk on the comb

Matthew Junge; Josh Meisel; Aldo Morelli

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arXiv · 2508.14977

Activated random walk on the comb

Matthew Junge·Josh Meisel·Aldo Morelli

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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.

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Matthew Junge, Josh Meisel, Aldo Morelli. 2025-08-20. Activated random walk on the comb. https://arxiv.org/abs/2508.14977

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