Competing frogs on ℤ^d

2019/02/05 by Deijfen, Maria, Hirscher, Timo, Lopes, Fabio
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1902.01849

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. We show that the event G_i that type i activates infinitely many particles has positive probability for all p₁,p₂∈(0,1] (i=1,2). Furthermore, if p₁=p₂, then the types can coexist in the sense that ℙ(G₁∩ G₂)>0. 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 p₁≠ p₂.

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