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

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 1s to the left of the origin and 0s elsewhere, we study the relaxation time $\tau$, that is, the first hitting time of the ground-state configuration (up to translation). In particular, 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 {\em et al.} [{\em Bernoulli} {\bf 7} (2001) 119--144]. Ours are the first non-existence of moments results for $\tau$ for the mixture model. Moreover, we give almost-sure asymptotic results on the long-term 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 the pure exclusion case. We state several significant open problems that remain.