Contact Interactions on a Lattice

Let \ξₜ\ be a Markov process whose values are subsets of Zd, the d-dimensional integers. Put ξₜ(x) = 1 if x ∈ ξₜ and 0 otherwise. The transition intensity for a change in ξₜ(x) depends on \ξₜ(y), y a neighbor of x\. The chief concern is with "contact processes," where ξₜ(x) can change from 0 to 1 only if ξₜ(y) = 1 for some y neighboring x. Let pₜ(ξ) = Prob \ξₜ ≠ ξ₀ = ξ\. Under appropriate conditions, pₜ is increasing, subadditive, or submodular in ξ. In the case of contact processes, conditions are giving implying that p_∞(ξ) = 0 for all finite ξ, or that the contrary is true. In other cases conditions for ergodicity are given.