Zero-temperature Glauber dynamics on the 3-regular tree and the median process

Michael Damron, Arnab Sen

Research output: Contribution to journalArticlepeer-review

Abstract

In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d. initial spins σx(0) from { - 1 , + 1 } with Ppx(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 T3, where it is known that the critical threshold pc, below which Pp-a.s. all spins fixate to - 1 , is strictly less than 1/2. Defining θ(p) to be the Pp-probability that a vertex fixates to + 1 , we show that θ is a continuous function on [0, 1], so that, in particular, θ(pc) = 0. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures Pp. 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 Pp 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 T3 in finite time.

Original languageEnglish (US)
Pages (from-to)25-68
Number of pages44
JournalProbability Theory and Related Fields
Volume178
Issue number1-2
DOIs
StatePublished - 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

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