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Communications in Mathematical Physics

, Volume 362, Issue 1, pp 185–217 | Cite as

Coarsening Model on \({\mathbb{Z}^{d}}\) with Biased Zero-Energy Flips and an Exponential Large Deviation Bound for ASEP

  • Michael Damron
  • Leonid Petrov
  • David Sivakoff
Article
  • 28 Downloads

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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Notes

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.

References

  1. AL.
    Aizenman M., Lebowitz J.L.: Metastability effects in bootstrap percolation. J. Phys. A Math. Gen. 21, 3801–3813 (1988)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. Arr.
    Arratia R.: Site recurrence for annihilating random walks on Z d. Ann. Probab. 11, 706–713 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BCOTT.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. BPP.
    Balogh J., Peres Y., Pete G.: Bootstrap percolation on infinite trees and nonamenable groups. Comb. Probab. Comput. 15, 715–730 (2006)CrossRefzbMATHGoogle Scholar
  5. Bor.
    Bornemann F.: On the numerical evaluation of Fredholm determinants. Math. Comput. 79(270), 871–915 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. CMST.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. CRL.
    Chalupa J., Reich G.R., Leath P.L.: Bootstrap percolation on a Bethe lattice. J. Phys. C 12, L31–L35 (1979)CrossRefGoogle Scholar
  8. DEKMS.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. FSS.
    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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. Joh.
    Johansson K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. Lac1.
    Lacoin H.: Approximate Lifshitz law for the zero-temperature stochastic Ising model in any dimension. Commun. Math. Phys. 318, 291–305 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. Lac2.
    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)Google Scholar
  13. Lig.
    Liggett, T.: Interacting Particle Systems. [Reprint of the 1985 original.] Springer, Berlin (2005)Google Scholar
  14. Mor.
    Morris R.: Zero-temperature Glauber dynamics on \({\mathbb{Z}^d}\). Probab. Theory Relat. Fields 149, 417–434 (2011)CrossRefzbMATHGoogle Scholar
  15. NNS.
    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)Google Scholar
  16. OKR.
    Olejarz J., Krapivsky P.L., Redner S.: Zero-temperature relaxation of three-dimensional Ising ferromagnet. Phys. Rev. E 83, 051104-1–051104-11 (2011)ADSGoogle Scholar
  17. Rost.
    Rost H.: Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 58, 41–53 (1981)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. Sch.
    Schonmann R.H.: On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20, 174–193 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  19. SKR.
    Spirin V., Krapivsky P.L., Redner S.: Freezing in Ising ferromagnet. Phys. Rev. E 65, 016119-1–016119-9 (2001)ADSCrossRefGoogle Scholar
  20. TW.
    Tracy C., Widom H.: Asymptotics in ASEP with step initial condition. Commun. Math. Phys. 290, 129–154 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. vE.
    van Enter A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  3. 3.Institute for Information Transmission ProblemsMoscowRussia
  4. 4.Departments of Statistics and MathematicsThe Ohio State UniversityColumbusUSA

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