Edge Fluctuations for the One Dimensional Supercritical Contact Process

Abstract

We consider the one dimensional supercritical contact process with initial configurations having infinitely many particles to the left of the origin and only finitely many to its right. Starting from any such configuration, we first prove that in the limit as time goes to infinity the law of the process, as seen from the edge, converges to the invariant distribution constructed by Durrett [12]. We then prove a functional central limit theorem for the fluctuations of the edge around its average, showing that the corresponding diffusion coefficient is strictly positive. We finally characterize the space time structure of the system. In particular we prove that its distribution shifted in space by $\alpha t$ ($t$ denotes the time and $\alpha$ the drift of the edge) converges when $t$ goes to infinity to a $\frac{1}{2} - \frac{1}{2}$ mixture of the two extremal invariant measures for the contact process.