Abstract

A two-type version of the frog model on 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) boolean AND G(2)(c) and G(1)(c) boolean AND G(2) both have positive probability for all p(1), p(2) is an element of (0, 1]. Furthermore, if p(1) = p(2), then the types can coexist in the sense that the event G(1) boolean AND 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 not equal P-2.