## Abstract

In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d. initial spins σ_{x}(0) from { - 1 , + 1 } with P_{p}(σ_{x}(0) = + 1) = p, and they update their spins at the arrival times of i.i.d. Poisson processes to agree with a majority of their neighbors. We study this process on the 3-regular tree T_{3}, where it is known that the critical threshold p_{c}, below which P_{p}-a.s. all spins fixate to - 1 , is strictly less than 1/2. Defining θ(p) to be the P_{p}-probability that a vertex fixates to + 1 , we show that θ is a continuous function on [0, 1], so that, in particular, θ(p_{c}) = 0. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures P_{p}. Along the way, we study the time-infinity agreement clusters of the median process, show that they are a.s. finite, and deduce that all continuous spins flip finitely often. In the second half of the paper, we show a correlation decay statement for the discrete spins under P_{p} for a.e. value of p. The proof relies on finiteness of a vertex’s “trace” in the median process to derive a stability of discrete spins under finite resampling. Last, we use our methods to answer a question of Howard (J Appl Probab 37:736–747, 2000) on the emergence of spin chains in T_{3} in finite time.

Original language | English (US) |
---|---|

Pages (from-to) | 25-68 |

Number of pages | 44 |

Journal | Probability Theory and Related Fields |

Volume | 178 |

Issue number | 1-2 |

DOIs | |

State | Published - Oct 1 2020 |

### Bibliographical note

Publisher Copyright:© 2020, Springer-Verlag GmbH Germany, part of Springer Nature.

## Keywords

- Invariant percolation
- Majority vote model
- Mass transport principle
- Median process
- Zero-temperature Glauber dynamics