Asymptotic behavior of the critical density of activated random walk

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.