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Passage-time moments and hybrid zones for the exclusion-voter model

MacPhee, Iain; Menshikov, Mikhail; Volkov, Stanislav LU and Wade, Andrew (2010) In Bernoulli 16(4). p.1312-1342
Abstract
We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1’s to the left of the origin and 0’s elsewhere, we study the relaxation time τ, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for τ to be finite and for certain moments of τ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119–144). Ours are the first non-existence-of-moments results for τ for the mixture model. Moreover, we give almost sure asymptotics for the... (More)
We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1’s to the left of the origin and 0’s elsewhere, we study the relaxation time τ, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for τ to be finite and for certain moments of τ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119–144). Ours are the first non-existence-of-moments results for τ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
keywords
almost-sure bounds, exclusion process, hybrid zone, Lyapunov functions, passage-time moments, voter model
in
Bernoulli
volume
16
issue
4
pages
1312 - 1342
publisher
Chapman and Hall
external identifiers
  • scopus:79251547040
ISSN
1350-7265
DOI
10.3150/09-BEJ243
language
English
LU publication?
no
id
b6693c0f-1368-4602-a917-ab04df7cc0d9 (old id 4359856)
alternative location
http://projecteuclid.org/euclid.bj/1290092908
date added to LUP
2014-03-17 14:29:40
date last changed
2018-06-10 03:10:17
@article{b6693c0f-1368-4602-a917-ab04df7cc0d9,
  abstract     = {We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and the exclusion process. With the process started from a finite perturbation of the ground state Heaviside configuration consisting of 1’s to the left of the origin and 0’s elsewhere, we study the relaxation time τ, that is, the first hitting time of the ground state configuration (up to translation). We give conditions for τ to be finite and for certain moments of τ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. (Bernoulli 7 (2001) 119–144). Ours are the first non-existence-of-moments results for τ for the mixture model. Moreover, we give almost sure asymptotics for the evolution of the size of the hybrid (disordered) region. Most of our results pertain to the discrete-time setting, but several transfer to continuous-time. As well as the mixture process, some of our results also cover pure exclusion. We state several significant open problems.},
  author       = {MacPhee, Iain and Menshikov, Mikhail and Volkov, Stanislav and Wade, Andrew},
  issn         = {1350-7265},
  keyword      = {almost-sure bounds,exclusion process,hybrid zone,Lyapunov functions,passage-time moments,voter model},
  language     = {eng},
  number       = {4},
  pages        = {1312--1342},
  publisher    = {Chapman and Hall},
  series       = {Bernoulli},
  title        = {Passage-time moments and hybrid zones for the exclusion-voter model},
  url          = {http://dx.doi.org/10.3150/09-BEJ243},
  volume       = {16},
  year         = {2010},
}