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

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

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.

Citation

MacPhee, I. M., Menshikov, M. V., Volkov, S., & Wade, A. R. (2010). Passage-time moments and hybrid zones for the exclusion-voter model. Bernoulli (Andover), 16(4), 1312-1342. https://doi.org/10.3150/09-bej243

• Journal Article Type: Article
• Publication Date: Nov 1, 2010
• Deposit Date: Feb 28, 2011
• Publicly Available Date: Jan 31, 2013
• Journal: Bernoulli - Journal of the Bernoulli Society
• Print ISSN: 1350-7265
• Electronic ISSN: 1573-9759
• Publisher: Bernoulli Society for Mathematical Statistics and Probability
• Peer Reviewed: Peer Reviewed
• Volume: 16
• Issue: 4
• Pages: 1312-1342
• DOI: https://doi.org/10.3150/09-bej243
• Keywords: Almost-sure bounds, Exclusion process, Hybrid zone, Lyapunov functions, Passage-time moments, Voter model.
• Public URL: https://durham-repository.worktribe.com/output/1534143