Correlation equations for the stable measure of a Markov chain

Each lamp of an infinite garland lights up with probability 1 if it and its neighbour both were lighting at the previous time moment, and with probability $\theta$ in the other case. It is shown that except for the trivial stable state “all the lamps are lighting”, for small $\theta$ there is only one stable probability measure $P_\theta$ on the state space of such systems and $P_\theta$ depends analitically on $\theta$.