## 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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## References

- 1.
Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab.

**12**(54), 1454–1508 (2007) - 2.
Arratia, R.: Site recurrence for annihilating random walks on \({\mathbb{Z}}^d\). Ann. Probab.

**11**, 706–713 (1983) - 3.
Backhausz, A., Szegedy, B., Virág, B.: Ramanujan graphings and correlation decay in local algorithms. Random Struct. Algorithms

**47**(3), 424–435 (2014) - 4.
Benjamini, I., Lyons, R., Peres, Y., Schramm, O.: Critical percolation on any nonamenable group has no infinite clusters. Ann. Probab.

**27**, 1347–1356 (1999) - 5.
Camia, F., Newman, C.M., Sidoravicius, V.: Approach to fixation for zero-temperature stochastic Ising models on the hexagonal lattice. In: Sidoravicius, V. (ed.) In and Out of Equilibrium. Progress in Probability, vol. 51. Birkhäuser, Boston (2002)

- 6.
Caputo, P., Martinelli, F.: Phase ordering after a deep quench: the stochastic Ising and hard core gas models on a tree. Probab. Theory Relat. Fields

**136**, 37–80 (2006) - 7.
Fontes, L., Schonman, R., Sidoravicius, V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Commun. Math. Phys.

**228**, 495–518 (2002) - 8.
Häggström, O.: Infinite clusters in dependent automorphism invariant percolation on trees. Ann. Probab.

**25**, 1423–1436 (1997) - 9.
Häggström, O.: Invariant percolation on trees and the mass-transport method. In: Bulletin of the International Statistical Institute, 52nd Session Proceedings. Tome LVIII, Book, vol. 1, pp. 363–366 (1999)

- 10.
Häggström, O.: Percolation beyond \({\mathbb{Z}}^d\): the contributions of Oded Schramm. Ann. Probab.

**39**, 1668–1701 (2011) - 11.
Harris, T.: A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab.

**5**, 451–454 (1977) - 12.
Howard, C.D.: Zero-temperature Ising spin dynamics on the homogeneous tree of degree three. J. Appl. Probab.

**37**, 736–747 (2000) - 13.
Kanoria, Y., Montanari, A.: Majority dynamics on trees and the dynamic cavity method. Ann. Appl. Probab.

**21**, 1694–1748 (2011) - 14.
Liggett, T.: Interacting Particle Systems. Reprint of the 1985 Original. Classics in Mathematics. Springer, Berlin (2005)

- 15.
Morris, R.: Zero-temperature Glauber dynamics on \({\mathbb{Z}}^d\). Probab. Theory Relat. Fields

**149**, 417–434 (2011) - 16.
Nanda, S., Newman, C.M., Stein, D.L.: Dynamics of Ising spin systems at zero temperature. In: Minlos, R., Shlosman, S., Suhov, Y. (eds.) On Dobrushin’s Way (from Probability Theory to Statistical Physics). American Mathematical Society, Providence (2000)

- 17.
Pete, G.: Probability and geometry on groups. Lecture notes available at http://math.bme.hu/~gabor/PGG.pdf (2019). Accessed Apr 2019

- 18.
Spirin, V., Krapivsky, P.L., Redner, S.: Freezing in Ising ferromagnet. Phys. Rev. E

**65**, 016119-1–016119-9 (2001) - 19.
Stauffer, D.: Ising spinodal decomposition at \(T = 0\) in one to five dimensions. J. Phys. A

**27**, 5029–55032 (1994) - 20.
Tessler, R.: Geometry and dynamics in zero temperature statistical mechanics models. arXiv: 1008.5279 (2010)

## Acknowledgements

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

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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,

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,

Consequently, for some large fixed time *T*,

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