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

TL;DR

This paper investigates conditions under which the continuous-time frog model on the integer lattice either explodes or does not, based on the initial distribution of particles and their activation dynamics.

Contribution

It provides new criteria for explosion and non-explosion in the continuous-time frog model, including cases with specific distributions of initial particles.

Findings

No explosion occurs if initial particles follow a distribution related to $e^{Y \, \ln Y}$ with finite expectation of Y.

Explosion occurs almost surely for distributions of $e^X$ with heavy tails, specifically when tail probability decays as $t^{-a}$ with $a \in (0,1)$.

Comparison to a percolation model is used to establish the results.

Abstract

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 $\mu$, $\mu(\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. 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.