A contact process with a single inhomogeneous site

Abstract

The one-dimensional basic contact process is a Markov process for which particles give birth on vacant nearest neighbor sites at rate λ>0 and particles die at rate one. We introduce a one-dimensional contact process with a single inhomogeneous site: the evolution is as above except that a particle located at the origin does not die. Let λ _c be the critical value of the basic contact process. We show that for λ≠λ _c the upper invariant measures of the inhomogeneous contact process and the basic contact process coincide except at a finite number of sites. The behavior at λ=λ _c is much more intersting: the upper invariant measure of the inhomogeneous contact process concentrates on configurations with infinitely many particles, while it is known that the critical basic contact process dies out. So a single inhomogeneity may provoke a perturbation unbounded in space. As a byproduct of our analysis we prove that the connectivity probabilities of the critical basic contact process are not summable. We also give a biological interpretation of this model.