## Abstract

We study the coarsening model (zero-temperature Ising Glauber dynamics) on \({\mathbb{Z}^{d}}\) (for \({d \geq 2}\)) with an asymmetric tie-breaking rule. This is a Markov process on the state space \({\{-1,+1\}^{{\mathbb{Z}}^d}}\) of “spin configurations” in which each vertex updates its spin to agree with a majority of its neighbors at the arrival times of a Poisson process. If a vertex has equally many +1 and −1 neighbors, then it updates its spin value to +1 with probability \({q \in [0,1]}\) and to −1 with probability 1 − *q*. The initial state of this Markov chain is distributed according to a product measure with probability *p* for a spin to be +1. In this paper, we show that for any given \({p > 0}\), there exist *q* close enough to 1 such that a.s. every spin has a limit of +1. This is of particular interest for small values of *p*, for which it is known that if \({q = 1/2}\), a.s. all spins have a limit of −1. For dimension *d* = 2, we also obtain near-exponential convergence rates for *q* sufficiently large, and for general *d*, we obtain stretched exponential rates independent of *d*. Two important ingredients in our proofs are refinements of block arguments of Fontes–Schonmann–Sidoravicius and a novel exponential large deviation bound for the Asymmetric Simple Exclusion Process.

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## References

Aizenman M., Lebowitz J.L.: Metastability effects in bootstrap percolation. J. Phys. A Math. Gen.

**21**, 3801–3813 (1988)Arratia R.: Site recurrence for annihilating random walks on

*Z*^{d}. Ann. Probab.**11**, 706–713 (1983)Benjamini I., Chan S.-O., O’Donnell R., Tamuz O., Tan L.-Y.: Convergence, unanimity, and disagreement in majority dynamics on unimodular graphs and random graphs. Stoch. Process. Appl.

**126**, 2719–2733 (2016)Balogh J., Peres Y., Pete G.: Bootstrap percolation on infinite trees and nonamenable groups. Comb. Probab. Comput.

**15**, 715–730 (2006)Bornemann F.: On the numerical evaluation of Fredholm determinants. Math. Comput.

**79**(270), 871–915 (2010)Caputo P., Martinelli F., Simenhaus F., Toninelli F.L.: “Zero” temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion. Commun. Pure Appl. Math.

**64**, 0778–0831 (2011)Chalupa J., Reich G.R., Leath P.L.: Bootstrap percolation on a Bethe lattice. J. Phys. C

**12**, L31–L35 (1979)Damron M., Eckner S.M., Kogan H., Newman C.M., Sidoravicius V.: Coarsening dynamics on \({\mathbb{Z}^d}\) with frozen vertices. J. Stat. Phys.

**160**, 60–72 (2015)Fontes L.R., Schonmann R.H., Sidoravicius V.: Stretched exponential fixation in stochastic Ising models at zero temperature. Commun. Math. Phys.

**228**, 495–518 (2002)Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys.

**209**(2), 437–476 (2000)Lacoin H.: Approximate Lifshitz law for the zero-temperature stochastic Ising model in any dimension. Commun. Math. Phys.

**318**, 291–305 (2013)Lacoin, H.: The scaling limit for zero-temperature planar Ising droplets: with and without magnetic fields. In: Topics in Percolative and Disordered Systems, Springer Proceedings in Mathematics and Statistics, vol. 69, pp. 85–120 (2014)

Liggett, T.: Interacting Particle Systems. [Reprint of the 1985 original.] Springer, Berlin (2005)

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

**149**, 417–434 (2011)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 Mechnics). American Mathematical Society Translations, Series II, vol. 198, pp. 183–193 (2000)

Olejarz J., Krapivsky P.L., Redner S.: Zero-temperature relaxation of three-dimensional Ising ferromagnet. Phys. Rev. E

**83**, 051104-1–051104-11 (2011)Rost H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete

**58**, 41–53 (1981)Schonmann R.H.: On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab.

**20**, 174–193 (1992)Spirin V., Krapivsky P.L., Redner S.: Freezing in Ising ferromagnet. Phys. Rev. E

**65**, 016119-1–016119-9 (2001)Tracy C., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys.

**290**, 129–154 (2009)van Enter A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys.

**48**, 943–945 (1987)

## Acknowledgments

MD thanks Eric Vigoda and Antonio Blanca for discussions related to the erosion times of boxes. MD and DS are grateful to Rob Morris for lengthy email discussions of coarsening dynamics. LP thanks Timo Seppäläinen and Benedek Valkó for helpful discussions on large deviation estimates in particle systems, and Ivan Corwin for useful remarks. MD is supported by an NSF CAREER grant. LP is partially supported by NSF Grant DMS-1664617. DS is supported by NSF Grants DMS-1418265 and CCF-1740761. We are grateful to anonymous referees for valuable suggestions.

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Damron, M., Petrov, L. & Sivakoff, D. Coarsening Model on \({\mathbb{Z}^{d}}\) with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP.
*Commun. Math. Phys.* **362**, 185–217 (2018). https://doi.org/10.1007/s00220-018-3180-2

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DOI: https://doi.org/10.1007/s00220-018-3180-2