The majority vote process and other consensus processes on trees

Maury Bramson, Lawrence F Gray

Research output: Contribution to journalArticlepeer-review


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 − ε, 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 ε ∈ [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 G = Zn with n ≥ 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 Td with vertex degree d. For d ≥ 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.

Original languageEnglish (US)
Pages (from-to)169-198
Number of pages30
JournalAnnals of Applied Probability
Issue number1
StatePublished - Feb 1 2021

Bibliographical note

Funding Information:
Acknowledgment. The first author was partially supported through NSF Grant DMS-1203201.

Funding Information:
The first author was partially supported through NSF Grant DMS-

Publisher Copyright:
© Institute of Mathematical Statistics, 2021


  • Consensus processes
  • Convergence to equilibrium
  • Equilibria
  • Majority vote processes
  • Trees


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