The majority vote process and other consensus processes on trees

Abstract

The majority vote process was one of the first interacting particle systems to be investigated. It can be described briefly as follows. There are two possible opinions at each site of a graph G. At rate $1-\mathit{\epsilon }$, the opinion at a site aligns with the majority opinion at its neighboring sites and, at rate ε, the opinion at a site is randomized due to noise, where $\mathit{\epsilon }\in [0,1]$ is a parameter.

Despite the simple dynamics of the majority vote process, its equilibrium behavior is difficult to analyze when the noise rate is small but positive. In particular, when the underlying graph is $\mathit{G}={\mathbb{Z}}^{\mathit{n}}$ with $\mathit{n}\ge 2$, it is not known whether the process possesses more than one equilibrium. This is surprising, especially in light of the close analogy between this model and the stochastic Ising model, where much more is known.

Here, we study the majority vote process on the infinite tree ${\mathbb{T}}_{\mathit{d}}$ with vertex degree d. For $\mathit{d}\ge 5$ and small noise, we show that there are uncountably many mutually singular equilibria, with convergence to such an equilibrium occurring exponentially quickly from nearby initial states.

Our methods are quite flexible and extend to a broader class of models, consensus processes. This class includes the stochastic Ising model and other processes in which the dynamics at a site depend on the number of neighbors holding a given opinion. All of our proofs are carried out in this broader context.