On a locally finite graph, consider the following interacting particle system, known as
Activate Random Walk (ARW). Start with a mass density $\mu$ of initially active particles, each of which performs a continuous time nearest neighbour symmetric random walk at rate one and falls asleep at rate $\lambda>0$. Sleepy particles become active on coming in contact with active particles.I shall describe two recent works on this model: one on the infinite lattice $\mathbb{Z}$, the second on the finite periodic lattice $\mathbb{Z}/n\mathbb{Z}$. On \mathbb{Z}, Rolla and Sidoravicius (Invent. Math., 2012) recently showed that for small enough particle density, almost surely the number of jumps at each site is finite. We complement the Rolla-Sidoravicius result by confirming a further physics prediction establishing that the critical density goes to zero along with the sleep rate. On the n-cycle, if the total number of particles is no more than n, almost surely the process reaches an absorbing state. We show a parallel quantitative phase transition by showing that the total number of jumps until absorption scales linearly in $n$ if $\mu$ is sufficiently small compared to $\lambda$ and exponentially in $n$ if $\lambda$ is sufficiently small compared to $\mu$.

Based on joint works in Shirshendu Ganguly, Christopher Hoffman and Jacob Richey.