2022/11/02 by Piet Lammers, Lammers, Piet, Fabio Lucio Toninelli +1
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2211.01033
openalex publication_date 2022/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
We study three Markov processes on infinite, unrooted, regular trees: the stochastic Ising model (also known as 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 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 looks nothing like a low-temperature 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 Zd, 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.