Transport and localization in random media: theory and applications

Majority vote processes on trees

Maury Bramson

University of Minnesota


The majority vote process was one of the first interacting particle systems to be investigated and can be described as follows. There are two possible opinions at each site and that opinion switches randomly to the majority opinion of the neighboring sites. Also, at a different rate $\varepsilon$, the opinion at each site randomly changes due to noise. Despite its simple dynamics, the majority vote process is difficult to analyze. In particular, on $\Z^d$ with $d>1$ and $\varepsilon$ chosen small, it is not known whether there exists more than one equilibrium. This is surprising due to the close analogy between the majority vote process and the Ising model. Here, we discuss work with Larry Gray on the majority vote process on the infinite tree with vertex degree $d$, where it is shown that, for small noise, there are uncountably many mutually singular equilibria, and that convergence to equilibrium occurs exponentially quickly from nearby initial states. Our methods rely on graphical constructions; they are quite flexible and can be used to obtain analogous results for other models, such as the stochastic Ising model on a tree.