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Competing frogs on zd

Deijfen, Maria ; Hirscher, Timo LU orcid and Lopes, Fabio (2019) In Electronic Journal of Probability 24.
Abstract

A two-type version of the frog model on Zd 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 Gi denote the event that type i activates... (More)

A two-type version of the frog model on Zd 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 Gi denote the event that type i activates infinitely many particles. We show that the events G1 ∩ Gc2 and Gc1 ∩ 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.

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Please use this url to cite or link to this publication:
author
; and
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Coexistence, Competing growth, Frog model, Random walk
in
Electronic Journal of Probability
volume
24
article number
146
publisher
UNIV WASHINGTON, DEPT MATHEMATICS
external identifiers
  • scopus:85077884991
ISSN
1083-6489
DOI
10.1214/19-EJP400
language
English
LU publication?
no
additional info
Publisher Copyright: © 2019, Institute of Mathematical Statistics. All rights reserved.
id
3befa6ff-49da-4208-94c2-c61120b3414c
date added to LUP
2023-12-14 13:22:17
date last changed
2025-10-14 11:16:26
@article{3befa6ff-49da-4208-94c2-c61120b3414c,
  abstract     = {{<p>A two-type version of the frog model on Z<sup>d</sup> is formulated, where active type i particles move according to lazy random walks with probability p<sub>i</sub> 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<sub>i</sub> denote the event that type i activates infinitely many particles. We show that the events G<sub>1</sub> ∩ G<sup>c</sup>2 and G<sup>c</sup>1 ∩ G<sub>2</sub> both have positive probability for all p<sub>1</sub>, p<sub>2</sub> ∈ (0, 1]. Furthermore, if p<sub>1</sub> = p<sub>2</sub>, then the types can coexist in the sense that the event G<sub>1</sub> ∩ G<sub>2</sub> 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<sub>1</sub> ≠ p<sub>2</sub>.</p>}},
  author       = {{Deijfen, Maria and Hirscher, Timo and Lopes, Fabio}},
  issn         = {{1083-6489}},
  keywords     = {{Coexistence; Competing growth; Frog model; Random walk}},
  language     = {{eng}},
  publisher    = {{UNIV WASHINGTON, DEPT MATHEMATICS}},
  series       = {{Electronic Journal of Probability}},
  title        = {{Competing frogs on z<sup>d</sup>}},
  url          = {{http://dx.doi.org/10.1214/19-EJP400}},
  doi          = {{10.1214/19-EJP400}},
  volume       = {{24}},
  year         = {{2019}},
}