Explosion and non-explosion for the continuous-time frog model

We consider the continuous-time frog model on $\mathbb{Z}$. At time $t=0$, there are $\eta(x)$ particles at $x \in \mathbb{Z}$, each of which is represented by a random variable. In particular, $(\eta(x))_{x \in \mathbb{Z}}$ is a collection of independent random variables with a common distribution $\mu, \mu\left(\mathbb{Z}_{+}\right)=1, \mathbb{Z}_{+}:=\mathbb{N} \cup\{0\}$, $\mathbb{N}=\{1,2,3, \ldots\}$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping, hence not active. Active particles perform a simple symmetric continuous-time random walk in $\mathbb{Z}$ (that is, a random walk with $\exp (1)$-distributed jump times and jumps -1 and 1 , each with probability $1 / 2$ ), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $\mu$ is the distribution of $e^{Y \ln Y}$ with a non-negative random variable $Y$ satisfying $\mathbb{E} Y<\infty$, then a.s. no explosion occurs. On the other hand, if $a \in(0,1)$ and $\mu$ is the distribution of $e^{X}$, where $\mathbb{P}\{X \geq t\}=t^{-a}, t \geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation.