Analyticity of the density and exponential decay of correlations in 2-d bootstrap percolation

We consider some deterministic cellular automata on the state space {0, 1}Zd, starting from the product of Bernoulli measures and evolving in discrete time according to the bootstrap percolation rules, in which a 0 changes to a 1 when it has at least l neighbours which are 1. We prove that in case l = 2d − 1 the limiting measure has an exponential decay of correlations and the density function is analytic in [0, 1].