Asymptotics of Locally Interacting Markov Chains with Global Signals

We study the long run behaviour of interactive Markov chains on in nite product spaces. The behaviour at a single site is in uenced by the local situation in some neighborhood and by a random signal about the average situation throughout the whole system. The asymp-totic behaviour of such Markov chains is analyzed on the microscopic level and on the macroscopic level of empirical elds. We give suÆ-cient conditions for convergence on the macroscopic level. Combining a convergence result from the theory of random systems with complete connections with a perturbation of the Dobrushin-Vasserstein contrac-tion technique we show that macroscopic convergence implies that the underlying Microscopic process has local asymptotic loss of memory. Key Words: Markov chains on innite product spaces, local asymptotic loss of memory, contraction techniques, Gibbs measure