## Electronic Communications in Probability

### Coarsening with a frozen vertex

#### 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
Accepted: 26 January 2016
First available in Project Euclid: 15 February 2016

https://projecteuclid.org/euclid.ecp/1455560033

Digital Object Identifier
doi:10.1214/16-ECP4785

Mathematical Reviews number (MathSciNet)
MR3485378

Zentralblatt MATH identifier
1336.60186

#### 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

#### 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.