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 of the system. In particular we prove that its distribution shifted in space by at (t denotes the time and a the drift of the edge) converges when t goes to infinity to a 2-2 mixture of the two extremal invariant measures for the contact process. 1. Introduction. In this article we study the one dimensional supercritical contact process, a stochastic process with infinitely many interacting particles which move on Z, the set of all the integers. We consider initial states with infinitely many particles to the left of the origin and finitely many to its right. We first prove that at large times the state of the system approaches some definite structure which, as time increases, rigidly shifts to the right: A steady state propagates from the left to the right with constant speed through the formerly empty region. The system is a (very schematic and rough) microscopic model for the formation and propagation of one dimensional shock waves, as we shall see in some more detail in the sequel. At large times we can distinguish three different space regions: The first two are semiinfinite intervals which extend to - oo and + oo, respectively, while the third one is the interval which connects the first two. In the first region the state looks like the nonzero density equilibrium state for the supercritical contact process, while, in the second it is the empty state. The third is the interesting region. It describes the front. We find that the state there is (close to) a superposition of the two extremal equilibrium states. The weight of the decomposition ranges from 1 (all the weight to the nonzero density state) to 0, as one moves from left to right. As the time t increases, the previous picture shifts to the right and, at the same time, the size of the front wave increases (like tl/2). The picture is very similar indeed to that found by Wick [25] in a zero range model. Our result is a consequence of a very detailed analysis of the motion of the first particle in the system, the edge, and of a precise knowledge of the of the state as seen from the edge. Such
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