Competing frogs on zd
(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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- author
- Deijfen, Maria ; Hirscher, Timo LU and Lopes, Fabio
- publishing date
- 2019
- 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
- 2023-12-14 15:55:09
@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}}, }