## The Annals of Applied Probability

### Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model

#### Abstract

We analyze clustering and (local) recurrence of a standard Markov process model of spatial domain coarsening. The continuous time process, whose state space consists of assignments of $+1$ or $-1$ to each site in $\mathbf{Z}^2$, is the zero-temperature limit of the stochastic homogeneous Ising ferromagnet (with Glauber dynamics): the initial state is chosen uniformly at random and then each site, at rate 1, polls its four neighbors and makes sure it agrees with the majority, or tosses a fair coin in case of a tie. Among the main results (almost sure, with respect to both the process and initial state) are: clusters (maximal domains of constant sign) are finite for times $t< \infty$, but the cluster of a fixed site diverges (in diameter) as $t \to \infty$; each of the two constant states is (positive) recurrent. We also present other results and conjectures concerning positive and null recurrence and the role of absorbing states.

#### Article information

Source
Ann. Appl. Probab., Volume 12, Number 2 (2002), 565-580.

Dates
First available in Project Euclid: 17 July 2002

https://projecteuclid.org/euclid.aoap/1026915616

Digital Object Identifier
doi:10.1214/aoap/1026915616

Mathematical Reviews number (MathSciNet)
MR1910640

Zentralblatt MATH identifier
1020.60094

#### Citation

Camia, Federico; De Santis, Emilio; Newman, Charles M. Clusters and recurrence in the two-dimensional zero-temperature stochastic ising model. Ann. Appl. Probab. 12 (2002), no. 2, 565--580. doi:10.1214/aoap/1026915616. https://projecteuclid.org/euclid.aoap/1026915616

#### References

• [1] ARRATIA, R. (1983). Site recurrence for annihilating random walks on Zd. Ann. Probab. 11 706-713.
• [2] BRAY, A. J. (1994). Theory of phase-ordering kinetics. Adv. Phy s. 43 357-459.
• [3] COX, J. T. and GRIFFEATH, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14 347-370.
• [4] COX, J. T. and KLENKE, A. (2000). Recurrence and ergodicity of interacting particle sy stems. Probab. Theory Related Fields 116 239-255.
• [5] DERRIDA, B. (1995). Exponents appearing in the zero-temperature dy namics of the 1D Potts model. J. Phy s. A 28 1481-1491.
• [6] DERRIDA, B., HAKIM, V. and PASQUIER, V. (1995). Exact first-passage exponents of 1D domain growth: Relation to a reaction-diffusion model. Phy s. Rev. Lett. 75 751-754.
• [7] DURRETT, R. (1988). Lecture Notes on Particle Sy stems and Percolation. Wadsworth and Brooks/Cole, Advanced Books and Software, Pacific Grove, CA.
• [8] FONTES, L. R., ISOPI, M. and NEWMAN, C. M. (1999). Chaotic time dependence in a disordered spin sy stem. Probab. Theory Related Fields 115 417-443.
• [9] FONTES, L. R., ISOPI, M. and NEWMAN, C. M. (2000). Random walks with strongly inhomogeneous rates and singular diffusion: Convergence, localization and aging in one dimension. Ann. Probab. 30 579-604.
• [10] FONTES, L. R., SIDORAVICIUS, V. and SCHONMANN, R. H. (2001). Stretched exponential fixation in stochastic Ising models at zero temperature. Preprint mp-arc 01-287.
• [11] GANDOLFI, A., KEANE, M. and RUSSO, L. (1988). On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation. Ann. Probab. 16 1147-1157.
• [12] GANDOLFI, A., NEWMAN, C. M. and STEIN, D. L. (2000). Zero-temperature dy namics of ±J spin glasses and related models. Comm. Math. Phy s. 214 373-387.
• [13] HARRIS, T. E. (1977). A correlation inequality for Markov processes in partially ordered state spaces. Ann. Probab. 5 451-454.
• [14] HOLLEY, R. A. and LIGGETT, T. M. (1975). Ergodic theorems for weakly interacting sy stems and the voter model. Ann. Probab. 3 643-663.
• [15] HOWARD, C. D. (2000). Zero-temperature Ising spin dy namics on the homogeneous tree of degree three. J. Appl. Probab. 37 736-747.
• [16] LIGGETT, T. M. (1985). Interacting Particle Sy stems. Springer, New York.
• [17] NANDA, S., NEWMAN, C. M. and STEIN, D. L. (2000). Dy namics of Ising spin sy stems at zero temperature. In On Dobrushin's Way ( from Probability Theory to Statistical Mechanics) (R. Minlos, S. Shlosman and Y. Suhov, eds.) 183-194. Amer. Math. Soc., Providence.
• [18] NEWMAN, C. M. and STEIN, D. L. (1999). Blocking and persistence in zero-temperature dy namics of homogeneous and disordered Ising models. Phy s. Rev. Lett. 82 3944-3947.
• [19] NEWMAN, C. M. and STEIN, D. L. (1999). Equilibrium pure states and nonequilibrium chaos. J. Statist. Phy s. 94 709-722.
• [20] NEWMAN, C. M. and STEIN, D. L. (2000). Zero-temperature dy namics of Ising spin sy stems following a deep quench: results and open problems. physica A 279 156-168.
• [21] STAUFFER, D. (1994). Ising spinodal decomposition at T = 0 in one to five dimensions. J. Phy s. A 27 5029-5032.
• NEW YORK, NY 10003 E-MAIL: federico.camia@physics.ny u.edu E. DE SANTIS DIPARTIMENTO DI MATEMATICA "GUIDO CASTELNUOVO" UNIVERSITÀ DI ROMA "LA SAPIENZA" PIAZZALE ALDO MORO 2 00185 ROMA ITALY E-MAIL: desantis@mat.uniroma1.it C. M. NEWMAN COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY
• NEW YORK, NY 10012 E-MAIL: newman@courant.ny u.edu