Competing frogs on Z^d

Artikel i vetenskaplig tidskrift, 2019

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 pi 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^c and G_1^c \cap G_2 both have positive probability for all 0< p_1, p_2 <=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 does not equal p_2.