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

Iain M. MacPhee, Mikhail V. Menshikov, Stanislav Volkov, Andrew Wade

Research output: Contribution to journalArticle

4 Citations (Scopus)

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.
LanguageEnglish
Pages1312-1342
Number of pages31
JournalBernoulli
Volume16
Issue number4
DOIs
Publication statusPublished - 2010

Fingerprint

Passage Time
Voter Model
Moment
Ground State
Oliver Heaviside
First Hitting Time
Nonequilibrium Dynamics
Exclusion Process
Interacting Particle Systems
Configuration
Mixture Model
Relaxation Time
Bernoulli
Nonexistence
Continuous Time
Open Problems
Discrete-time
Cover
Perturbation

Keywords

  • almost-sure bounds
  • exclusion process
  • hybrid zone
  • voter model
  • passage-time moments
  • lyapunov functions

Cite this

MacPhee, I. M., Menshikov, M. V., Volkov, S., & Wade, A. (2010). Passage-time moments and hybrid zones for the exclusion-voter model. Bernoulli, 16(4), 1312-1342. https://doi.org/10.3150/09-BEJ243
MacPhee, Iain M. ; Menshikov, Mikhail V. ; Volkov, Stanislav ; Wade, Andrew. / Passage-time moments and hybrid zones for the exclusion-voter model. In: Bernoulli. 2010 ; Vol. 16, No. 4. pp. 1312-1342.
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MacPhee, IM, Menshikov, MV, Volkov, S & Wade, A 2010, 'Passage-time moments and hybrid zones for the exclusion-voter model' Bernoulli, vol. 16, no. 4, pp. 1312-1342. https://doi.org/10.3150/09-BEJ243

Passage-time moments and hybrid zones for the exclusion-voter model. / MacPhee, Iain M.; Menshikov, Mikhail V.; Volkov, Stanislav; Wade, Andrew.

In: Bernoulli, Vol. 16, No. 4, 2010, p. 1312-1342.

Research output: Contribution to journalArticle

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