The contact process with dynamic edges on Z

Abstract

We study the contact process running in the one-dimensional lattice undergoing dynamical percolation, where edges open at rate vp and close at rate v(1-p). Our goal is to explore how the speed of the environment, v, affects the behavior of the process. Among our main results we find that: 1. For small enough v the process dies out, while for large v the process behaves like a contact process on Z with rate lambda p, where lambda is the birth rate of each particle, so in particular it survives if lambda is large. 2. For fixed v and small enough p the network becomes immune, in the sense that the process dies out for any infection rate lambda, while if p is sufficiently close to 1 then for all v > 0 survival is possible for large enough lambda. 3. Even though the first two points suggest that larger values of v favor survival, this is not necessarily the case for small v: when the number of initially infected sites is large enough, the infection survives for a larger expected time in a static environment than in the case of v positive but small. Some of these results hold also in the setting of general (infinite) vertex-transitive regular graphs.