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

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 Howard (J Appl Probab 37:736–747, 2000) on the emergence of spin chains in \({\mathbb {T}}_3\) in finite time.

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Acknowledgements

AS thanks M. Bramson, E. Mossel, and O. Tamuz for helpful discussions.

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Correspondence to Michael Damron.

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The research of MD is supported by an NSF CAREER Grant. The research of AS is partially supported by NSF DMS 1406247.

Appendix

Appendix

First we prove Lemma 1.12, which bounds the probability of existence of long chronological paths.

Proof of Lemma 1.12

If \(\Gamma \) is a deterministic path starting from o with \(\ell \) many vertices, the time it takes for successive clock rings to occur along \(\Gamma \) is at least \(\sum _{i=1}^\ell \tau _i\), where the \(\tau _i\)’s are i.i.d. exponential random variables with mean one. There are \(4^{\ell -1}\) many paths starting from or ending at o with \(\ell \) many vertices, so by a union bound and the Markov inequality,

$$\begin{aligned}&{\mathbb {P}}\ (\exists \text { chronological path with } \ell \text { many vertices starting from or ending at }o \text { for } [0,T]) \\&\quad \le 4^{\ell -1} {\mathbb {P}}\left( \sum _{i=1}^\ell \tau _i \le T\right) \\&\quad = 4^{\ell -1} {\mathbb {P}}\left( \exp \left( - 4 \sum _{i=1}^\ell \tau _i \right) \ge e^{-4T}\right) \\&\quad \le 4^{\ell -1} \frac{{\mathbb {E}}\exp \left( - 4 \sum _{i=1}^\ell \tau _i\right) }{e^{-4T}} = \frac{e^{4T}}{4} \left( \frac{4}{5} \right) ^\ell . \end{aligned}$$

Summing this bound over \(\ell \ge k\) gives the statement of the lemma. \(\square \)

Next, we state and prove a result showing that if \(q \in [0,1]\) has \(\theta (q)>0\), then with positive probability, in the majority vote model with initial bias q, the root starts with spin \(+1\) and never flips.

Lemma 7.1

Consider the majority dynamics on \({\mathbb {T}}_3\) with initial spin configuration distributed according to the i.i.d. product measure \(\mu _q\) with \(q \in [0,1]\) satisfying \(\theta (q) > 0\). Let us keep the spin one of the neighbors, say \(x_0\), of the root o frozen at \(-1\) for all time. Then with positive probability, the spin at the root is \(+1\) at \(t=0\) and it never flips. Clearly, this event depends only on the clocks and initial spins of the vertices in the subtree \({\mathbb {T}}_{o \rightarrow x_0}\).

Proof

Let us denote the three neighbors of o in \({\mathbb {T}}_3\) by \(x_{-1}, x_0, x_1\). By a result of Harris [11, 14], it follows that the measure \(\mu ^t\) on \(\{-1, 1\}^{ {\mathcal {V}}}\) describing the state \(\sigma (t)\) of the system at time \(t \in [0, \infty ]\) possesses the FKG property; i.e., increasing functions of the spin variables are positively correlated. In fact, this follows from the FKG property of \(\mu ^0\) (which holds trivially since \(\mu ^0\) is an i.i.d. measure) and the attractiveness of the Markov process. Therefore,

$$\begin{aligned} {\mathbb {P}}_q( \sigma _{x_{-1}}(\infty )= & {} +1, \sigma _{x_{1}}(\infty ) = +1 ) \ge {\mathbb {P}}_q( \sigma _{x_{-1}}(\infty ) = +1) {\mathbb {P}}_q( \sigma _{x_{1}}(\infty ) = +1 ) \\= & {} \theta ^2(q)> 0. \end{aligned}$$

Consequently, for some large fixed time T,

$$\begin{aligned} \text { the event } A:= \{ \sigma _{x_{-1}}(t) = +1, \sigma _{x_{1}}(t) = +1 \ \text { for all } t \ge T \} \ \text { has positive probability}. \end{aligned}$$

As before, let \(\sigma (0)\) be the discrete spins at \(t=0\) and \(\omega \) be a realization of the Poisson clocks of the vertices in \({\mathbb {T}}_3\). We define a modification operator \(\Psi : (\sigma (0), \omega ) \mapsto (\sigma '(0), \omega ')\) by setting the initial spin at o to be \(+1\) and by suppressing all clock rings of o in [0, T] so that the first ring of the clock at o happens after time T. Let \(A'\) be the event obtained from A after applying this modification, i.e., \(A' = \{ \Psi ((\sigma (0), \omega ) ): (\sigma (0), \omega ) \in A\}\). Then \(A'\) also has positive probability.

Since \(1= \sigma '_o(t) \ge \sigma _o(t)\) for all \(0 \le t \le T\), we claim that \(\sigma '_y(t) \ge \sigma _y(t)\) for all \(0 \le t \le T\) and for each vertex y. Indeed, at the time of each clock ring at any \(x \in \partial o\) in the interval [0, T], the spin of its neighbor o in \((\sigma '(0), \omega ')\) dominates that in \((\sigma (0), \omega )\). Therefore, we have \(\sigma '_{x}(t) \ge \sigma _{x}(t)\) for all \(0 \le t \le T\) and for any \(x \in \partial o\). Applying the argument iteratively to the vertices lying at distance \(r=1,2, \ldots \) from o yields the claim.

Since \(\sigma '_y(T) \ge \sigma _y(T)\) for all y and the clock rings at every vertex are identical in \(\omega \) and \(\omega '\) after T, it follows from the attractiveness property of the majority dynamics that \(\sigma '_x(t) \ge \sigma _x(t)\) for all x and for all \(t \ge T\). In particular, \(\sigma '_{x_{-1}}(t) = +1\) and \(\sigma '_{x_{1}}(t) = +1\) for all \( t \ge T\). Therefore, at the time of the each ring in \(\omega _o'\) (which, by definition, occurs after time T), the vertex o has at least two neighbors with \(+1\) spins. Hence, \(\sigma '_o(t) =+1\) for all time t after T. So, on the event \(A'\), the spin at the root is \(+1\) at \(t=0\) and it never flips. The lemma follows. \(\square \)

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Damron, M., Sen, A. Zero-temperature Glauber dynamics on the 3-regular tree and the median process. Probab. Theory Relat. Fields 178, 25–68 (2020). https://doi.org/10.1007/s00440-020-00968-9

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Keywords

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

Mathematics Subject Classification

  • 60K35
  • 82C22