Competing frogs on z^d

Summary, in English

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 G₁ ∩ G^c2 and G^c1 ∩ G₂ both have positive probability for all p₁, p₂ ∈ (0, 1]. Furthermore, if p₁ = p₂, then the types can coexist in the sense that the event G₁ ∩ G₂ 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 p₁ ≠ p₂.

• Author: Maria Deijfen; Timo Hirscher; Fabio Lopes
• Publishing year: 2019
• Language: English
• Publication/Series: Electronic Journal of Probability
• Volume: 24
• Links: http://dx.doi.org/10.1214/19-EJP400
• Document type: Journal article
• Publisher: UNIV WASHINGTON, DEPT MATHEMATICS
• Topic: Probability Theory and Statistics
• Keywords: Coexistence; Competing growth; Frog model; Random walk
• Status: Published
• ISBN/ISSN/Other: ISSN: 1083-6489