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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 isrepresented by a random variable. In particular, $(\eta(x))_{x \in \mathbb{Z}}$ is a collection of independent random variables with a common distribution$\mu$, $\mu(\mathbb{Z}_+) = 1$. The particles at the origin are active, allother ones being assumed as dormant, or sleeping. Active particles perform asimple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, arandom walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, eachwith probability $1/2$), independently of all other particles. Sleepingparticles stay still until the first arrival of an active particle to theirlocation; upon arrival they become active and start their own simple randomwalks. Different sets of conditions are given ensuring explosion, respectivelynon-explosion, of the continuous-time frog model. Our results show inparticular that if $\mu$ is the distribution of $e^{Y \ln Y}$ with anon-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 thedistribution 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 toa percolation model which we call totally asymmetric discrete inhomogeneousBoolean percolation.

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