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

Abstract

We consider the continuous-time frog model on Z. At time t=0, there are η(x) particles at x∈Z, each of which is represented by a random variable. In particular, (η(x))x∈Z is a collection of independent random variables with a common distribution μ, μ(Z+)=1, Z+≔N∪{0}, N={1,2,3,…}. 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 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 μ is the distribution of eYlnY with a non-negative random variable Y satisfying EY<∞, then a.s. no explosion occurs. On the other hand, if a∈(0,1) and μ is the distribution of eX, where P{X≥t}=t−a, t≥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.