Sharpness of the percolation transition for a class of contact processes - Part (I)

Motivated by models of vegetation patterns in the life-science literature, where a remarkable form of `robust critical behaviour' was suggested, we study percolation phenomena under the invariant measures of certain contact-like processes. We show that the percolation transition is sharp. (For one of the models with three states this is done under an assumption which will be explained in the talk). This sharpness seems to contradict the above mentioned robustness.  In Part (I) I will introduce the models, give some percolation background, and discuss weak-influence inequalities. Part (II) will present the above mentioned results, which were recently obtained with Jakob Bjornberg and Markus Heydenreich.