Electronic Communications in Probability

Coarsening with a frozen vertex

Michael Damron, Hana Kogan, Charles M. Newman, and Vladas Sidoravicius

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Abstract

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z} ^2}$ and initial state chosen from symmetric product measure, it is known (see [2]) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 9, 4 pp.

Dates
Received: 29 December 2015
Accepted: 26 January 2016
First available in Project Euclid: 15 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1455560033

Digital Object Identifier
doi:10.1214/16-ECP4785

Mathematical Reviews number (MathSciNet)
MR3485378

Zentralblatt MATH identifier
1336.60186

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C22: Interacting particle systems [See also 60K35]

Keywords
coarsening models zero-temperature Glauber dynamics frozen vertex

Rights
Creative Commons Attribution 4.0 International License.

Citation

Damron, Michael; Kogan, Hana; Newman, Charles M.; Sidoravicius, Vladas. Coarsening with a frozen vertex. Electron. Commun. Probab. 21 (2016), paper no. 9, 4 pp. doi:10.1214/16-ECP4785. https://projecteuclid.org/euclid.ecp/1455560033


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References

  • [1] M. Damron, S. M. Eckner, H. Kogan, C. M. Newman, V. Sidoravicius. Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices. J. Stat. Phys. 160, pp. 60-72, 2015.
  • [2] S. Nanda, C. M. Newman, D. L. Stein, Dynamics of Ising spins systems at zero temperature. In: On Dobrushin’s way (From Probability Theory to Statistical Mechanics). R. Milnos, S. Shlosman and Y. Suhov, eds., Am. Math. Soc. Transl. (2) 198, pp. 183–194, 2000.
  • [3] C. M. Newman, D. L. Stein, Zero-temperature dynamics of Ising spin systems following a deep quench: results and open problems. Physica A. 279, pp. 159–168, 2000.