Competing frogs on ${\mathbb Z}^{d}$

Abstract

A two-type version of the frog model on $\mathbb{Z} ^{d}$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_{i}$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. Let $G_{i}$ denote the event that type $i$ activates infinitely many particles. We show that the events $G_{1}\cap G_{2}^{\mathrm{c} }$ and $G_{1}^{\mathrm{c} }\cap G_{2}$ both have positive probability for all $p_{1},p_{2}\in (0,1]$. Furthermore, if $p_{1}=p_{2}$, then the types can coexist in the sense that the event $G_{1}\cap G_{2}$ has positive probability. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when $p_{1}\neq p_{2}$.