Asymptotic behavior of the critical density of activated random walk

TL;DR

This paper investigates how the critical density of activated random walks behaves asymptotically as the sleep rate varies, providing new bounds and approximations in different dimensions and regimes.

Contribution

It introduces new lower bounds and first-order approximations for the critical density in the regimes of large and small sleep rates, advancing understanding of the model's phase transition.

Findings

Critical density approaches 1 superpolynomially fast in 1D for large $mbda$.

New lower bounds for the decay rate of critical density in 2D for small $mbda$.

First-order approximation for transient walks in both regimes.

Abstract

We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $\lambda$ tends to $0$ and $\infty$. For large $\lambda$, we prove new lower bounds in dimensions 1 and 2, showing that in one dimension the critical density approaches $1$ superpolynomially fast. For small $\lambda$, we prove a new lower bound in two dimensions for how fast the critical density vanishes. We also obtain the first-order approximation for transient walks in both regimes.