Survival of Cyclical Particle Systems

We consider here a continuous time interacting particle system on Z with possible states 0,1,…,N-1 at each site. We denote by ξ_t(i) the state taken at site i and time t. Our models, which we call cyclical particle systems, are specified by the following transition rates: $${\xi _{\rm{t}}}{\rm{(i)}} \to {\xi _{\rm{t}}}({\rm{i}}) + {\rm{l (mod}}{\mkern 1mu} {\rm{N) at rate }}{\mkern 1mu} \lambda ,$$ where $$\lambda = \left| {\,\left\{ {\,{\rm{j}} = \pm 1:\,{\xi _{\rm{t}}}({\rm{i + j) = }}{\xi _{\rm{t}}}({\rm{i) + 1}}\,{\mkern 1mu} {\rm{(mod}}\;{\rm{N)}}} \right\}\,} \right|.$$. Note in particular that i cannot change state until at least one of its immediate neighbors has state ξ_t(i) + 1. For convenience, one may equip ξ with a space-time percolation substructure. Arrows from i to i+1 and i-1 are each put dow in a Poisson manner with rate 1. A state change is induced at (t, i) by an arrow at time t which enters i from i+1 or i-1, where the state is ξ_t(i) + 1. In diagram 1 below, numbers at the points of arrows indicate state change in the realization (N=5). Here, we will always assume that ξ₀ has product measure with $${\rm{P(}}{\xi _0}{\rm{(i) = k) = 1/N}}\;{\rm{for}}\;{\rm{k = 0,}}{\mkern 1mu} 1,...,{\mkern 1mu} {\rm{N - 1}}{\rm{.}}$$