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

TL;DR

This paper studies zero-temperature Glauber dynamics on the 3-regular tree, introducing a median process to prove the continuity of fixation probabilities and analyzing the decay of correlations.

Contribution

It introduces the median process to couple measures and proves the continuity of fixation probability, also analyzing agreement clusters and correlation decay.

Findings

The fixation probability function is continuous on [0,1].

Agreement clusters are almost surely finite.

Discrete spins flip finitely often.

Abstract

In zero-temperature Glauber dynamics, vertices of a graph are given i.i.d.~initial spins $\sigma_x(0)$ from $\{-1,+1\}$ with $\mathbb{P}_p(\sigma_x(0) = +1)=p$, 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 $\mathbb{T}_3$, where it is known that the critical threshold $p_c$, below which $\mathbb{P}_p$-a.s. all spins fixate to $-1$, is strictly less than $1/2$. Defining $\theta(p)$ to be the $\mathbb{P}_p$-probability that a vertex fixates to $+1$, we show that $\theta$ is a continuous function on $[0,1]$, so that, in particular, $\theta(p_c)=0$. To do this, we introduce a new continuous-spin process we call the median process, which gives a coupling of all the measures $\mathbb{P}_p$. 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 $\mathbb{P}_p$ 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 C. Howard (2001) on the emergence of spin chains in $\mathbb{T}_3$ in finite time.