The Annals of Probability

Sharp metastability threshold for an anisotropic bootstrap percolation model

H. Duminil-Copin and A. C. D. Van Enter

Full-text: Open access

Abstract

Bootstrap percolation models have been extensively studied during the two past decades. In this article, we study the following “anisotropic” bootstrap percolation model: the neighborhood of a point $(m,n)$ is the set

\[\{(m+2,n),(m+1,n),(m,n+1),(m-1,n),(m-2,n),(m,n-1)\}.\]

At time 0, sites are occupied with probability $p$. At each time step, sites that are occupied remain occupied, while sites that are not occupied become occupied if and only if three of more sites in their neighborhood are occupied. We prove that it exhibits a sharp metastability threshold. This is the first mathematical proof of a sharp threshold for an anisotropic bootstrap percolation model.

Article information

Source
Ann. Probab., Volume 41, Number 3A (2013), 1218-1242.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1367241499

Digital Object Identifier
doi:10.1214/11-AOP722

Mathematical Reviews number (MathSciNet)
MR3098677

Zentralblatt MATH identifier
1356.60166

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 83B43 83C43

Keywords
Bootstrap percolation sharp threshold anisotropy metastability

Citation

Duminil-Copin, H.; Van Enter, A. C. D. Sharp metastability threshold for an anisotropic bootstrap percolation model. Ann. Probab. 41 (2013), no. 3A, 1218--1242. doi:10.1214/11-AOP722. https://projecteuclid.org/euclid.aop/1367241499


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References

  • [1] Aizenman, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation. J. Phys. A 21 3801–3813.
  • [2] Amini, H. (2010). Bootstrap percolation in living neural networks. J. Stat. Phys. 141 459–475.
  • [3] Balogh, J., Bollobás, B., Duminil-Copin, H. and Morris, R. (2012). The sharp threshold for bootstrap percolation in all dimensions. Trans. Amer. Math. Soc. 364 2667–2701.
  • [4] Balogh, J., Bollobás, B. and Morris, R. (2009). Bootstrap percolation in three dimensions. Ann. Probab. 37 1329–1380.
  • [5] Balogh, J., Bollobás, B. and Morris, R. (2009). Majority bootstrap percolation on the hypercube. Combin. Probab. Comput. 18 17–51.
  • [6] Balogh, J., Bollobás, B. and Morris, R. (2010). Bootstrap percolation in high dimensions. Combin. Probab. Comput. 19 643–692.
  • [7] Cerf, R. and Cirillo, E. N. M. (1999). Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27 1837–1850.
  • [8] Cerf, R. and Manzo, F. (2002). The threshold regime of finite volume bootstrap percolation. Stochastic Process. Appl. 101 69–82.
  • [9] Chalupa, J., Leath, P. L. and Reich, G. R. (1979). Bootstrap percolation on a Bethe lattice. J. Phys. C 12 31–35.
  • [10] de Gregorio, P., Dawson, K. A. and Lawlor, A. (2009). Bootstrap percolation. In Springer Encyclopedia of Complexity and Systems Science 2 608–626. Springer, Berlin.
  • [11] Duarte, J. A. M. S. (1989). Simulation of a cellular automaton with an oriented bootstrap rule. Phys. A 157 1075–1079.
  • [12] Duminil-Copin, H. and Holroyd, A. (2012). Finite volume bootstrap percolation with threshold dynamics on Z2 I: Balanced case. Unpublished manuscript.
  • [13] Fey, A., Levine, L. and Peres, Y. (2010). Growth rates and explosions in sandpiles. J. Stat. Phys. 138 143–159.
  • [14] Garrahan, J. P., Sollich, P. and Toninelli, C. (2011). Kinetically constrained models. In Dynamical Heterogeneities in Glasses, Colloids, and Granular Media (L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti and W. van Saarloos, eds.). International Series of Monographs on Physics 150 111–137. Oxford Univ. Press, Oxford.
  • [15] Gravner, J. and Griffeath, D. (1996). First passage times for threshold growth dynamics on $\mathbb{Z}^{2}$. Ann. Probab. 24 1752–1778.
  • [16] Gravner, J. and Griffeath, D. (1999). Scaling laws for a class of critical cellular automaton growth rules. In Random Walks (Budapest, 1998). Bolyai Soc. Math. Stud. 9 167–186. János Bolyai Math. Soc., Budapest.
  • [17] Gravner, J., Holroyd, A. and Morris, R. (2012). A sharper threshold for bootstrap percolation in two dimensions. Probab. Theory Related Fields 153 1–23.
  • [18] Gravner, J. and McDonald, E. (1997). Bootstrap percolation in a polluted environment. J. Stat. Phys. 87 915–927.
  • [19] Grimmett, G. (1999). Percolation, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 321. Springer, Berlin.
  • [20] Holroyd, A. E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields 125 195–224.
  • [21] Holroyd, A. E. (2006). The metastability threshold for modified bootstrap percolation in $d$ dimensions. Electron. J. Probab. 11 418–433 (electronic).
  • [22] Holroyd, A. E., Liggett, T. M. and Romik, D. (2004). Integrals, partitions, and cellular automata. Trans. Amer. Math. Soc. 356 3349–3368 (electronic).
  • [23] Morris, R. (2010). The phase transition for bootstrap percolation in two dimensions. Unpublished manuscript.
  • [24] Mountford, T. S. (1993). Comparison of semi-oriented bootstrap percolation models with modified bootstrap percolation. In Cellular Automata and Cooperative Systems (Les Houches, 1992). NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 396 519–523. Kluwer Academic, Dordrecht.
  • [25] Mountford, T. S. (1995). Critical length for semi-oriented bootstrap percolation. Stochastic Process. Appl. 56 185–205.
  • [26] Schonmann, R. H. (1990). Finite size scaling behavior of a biased majority rule cellular automaton. Phys. A 167 619–627.
  • [27] Schonmann, R. H. (1990). Critical points of two-dimensional bootstrap percolation-like cellular automata. J. Stat. Phys. 58 1239–1244.
  • [28] Schonmann, R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20 174–193.
  • [29] Tlusty, T. and Eckmann, J. P. (2009). Remarks on bootstrap percolation in metric networks. J. Phys. A 42 205004.
  • [30] Toninelli, C. (2006). Bootstrap and jamming percolation. In Notes of Les Houches Summer School (J. P. Bouchaud, M. Mézard and J. Dalibard, eds.) 85 289–308. Elsevier, Berlin.
  • [31] van Enter, A. C. D. (1987). Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48 943–945.
  • [32] van Enter, A. C. D., Adler, J. and Duarte, J. A. M. S. (1990). Finite-size effects for some bootstrap percolation models. J. Stat. Phys. 60 323–332.
  • [33] van Enter, A. C. D. andFey, A. (2012). Metastability threshold for anisotropic bootstrap percolation in three dimensions. J. Stat. Phys. 147 97–112.
  • [34] van Enter, A. C. D. andHulshof, T. (2007). Finite-size effects for anisotropic bootstrap percolation: Logarithmic corrections. J. Stat. Phys. 128 1383–1389.