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We study three Markov processes on infinite, regular trees: the Glauber (heatbath) dynamics of the Ising model, a majority voter dynamic, and a coalescingparticle model. In each of the three cases the tree exhibits a preferreddirection which is encoded into the model. For all three models, our mainresult is the existence of a stationary but non-reversible measure. For theIsing model, this requires imposing that the inverse temperature is large andchoosing suitable non-uniform couplings, and our theorem implies the existenceof a stationary measure which does not at all look like a low-temperature IsingGibbs measure. The interesting aspect of our results lies in the fact that theanalogous processes do not have non-Gibbsian stationary measures on $\mathbbZ^d$, owing to the amenability of that graph. In fact, no example of astochastic Ising model with a non-reversible stationary state was known todate.

Non-reversible stationary states for majority voter and Ising dynamics on trees