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

We study three Markov processes on infinite, regular trees: the Glauber (heat bath) dynamics of the Ising model, a majority voter dynamic, and a coalescing particle model. In each of the three cases the tree exhibits a preferred direction which is encoded into the model. For all three models, our main result is the existence of a stationary but non-reversible measure. For the Ising model, this requires imposing that the inverse temperature is large and choosing suitable non-uniform couplings, and our theorem implies the existence of a stationary measure which does not at all look like a low-temperature Ising Gibbs measure. The interesting aspect of our results lies in the fact that the analogous processes do not have non-Gibbsian stationary measures on $\mathbb Z^d$, owing to the amenability of that graph. In fact, no example of a stochastic Ising model with a non-reversible stationary state was known to date.Comment: 13 page