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

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

Bernoulli, 16, no. 4, November 2010, 1312–1342. DOI: 10.3150/09-BEJ243

Abstract

We study the non-equilibrium dynamics of a one-dimensional interacting particle system that is a mixture of the voter model and 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 $\tau$, that is, the first hitting time of the ground-state configuration (up to translation). We give conditions for $\tau$ to be finite and for certain moments of $\tau$ to be finite or infinite, and prove a result that approaches a conjecture of Belitsky et al. [Bernoulli 7 (2001) 119-144; MR1811747]. Ours are the first non-existence of moments results for $\tau$ 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.