The Annals of Probability

The sharp threshold for the Duarte model

Béla Bollobás, Hugo Duminil-Copin, Robert Morris, and Paul Smith

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Abstract

The class of critical bootstrap percolation models in two dimensions was recently introduced by Bollobás, Smith and Uzzell, and the critical threshold for percolation was determined up to a constant factor for all such models by the authors of this paper. Here, we develop and refine the techniques introduced in that paper in order to determine a sharp threshold for the Duarte model. This resolves a question of Mountford from 1995, and is the first result of its type for a model with drift.

Article information

Source
Ann. Probab. Volume 45, Number 6B (2017), 4222-4272.

Dates
Received: March 2016
Revised: October 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1513069259

Digital Object Identifier
doi:10.1214/16-AOP1163

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60C05: Combinatorial probability

Keywords
Bootstrap percolation monotone cellular automata Duarte model critical probability sharp threshold

Citation

Bollobás, Béla; Duminil-Copin, Hugo; Morris, Robert; Smith, Paul. The sharp threshold for the Duarte model. Ann. Probab. 45 (2017), no. 6B, 4222--4272. doi:10.1214/16-AOP1163. https://projecteuclid.org/euclid.aop/1513069259


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References

  • [1] Aizenman, M. and Lebowitz, J. L. (1988). Metastability effects in bootstrap percolation.J. Phys. A213801–3813.
  • [2] Balister, P., Bollobás, B., Przykucki, M. and Smith, P. (2016). Subcritical $\mathcal{U}$-bootstrap percolation models have non-trivial phase transitions.Trans. Amer. Math. Soc.3687385–7411.
  • [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.3642667–2701.
  • [4] Balogh, J., Bollobás, B. and Morris, R. (2009). Bootstrap percolation in three dimensions.Ann. Probab.371329–1380.
  • [5] Bollobás, B., Duminil-Copin, H., Morris, R. and Smith, P. J. Universality of two-dimensional critical cellular automata. Preprint. Available atarXiv:1406.6680.
  • [6] Bollobás, B., Smith, P. and Uzzell, A. (2015). Monotone cellular automata in a random environment.Combin. Probab. Comput.24687–722.
  • [7] Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2008). Kinetically constrained spin models.Probab. Theory Related Fields140459–504.
  • [8] Cerf, R. and Cirillo, E. N. M. (1999). Finite size scaling in three-dimensional bootstrap percolation.Ann. Probab.271837–1850.
  • [9] Cerf, R. and Manzo, F. (2002). The threshold regime of finite volume bootstrap percolation.Stochastic Process. Appl.10169–82.
  • [10] Chalupa, J., Leath, P. L. and Reich, G. R. (1979). Bootstrap percolation on a Bethe lattice.J. Phys. C,Solid State Phys.12L31–L35.
  • [11] Duarte, J. A. M. S. (1989). Simulation of a cellular automaton with an oriented bootstrap rule.Phys. A1571075–1079.
  • [12] Duminil-Copin, H. and Holroyd, A. E. Finite volume bootstrap percolation with threshold rules on $\mathbb{Z}^{2}$: Balanced case. Preprint. Available athttp://www.unige.ch/~duminil/.
  • [13] Duminil-Copin, H. and van Enter, A. C. D. (2013). Sharp metastability threshold for an anisotropic bootstrap percolation model.Ann. Probab.411218–1242.
  • [14] Fontes, L. R., Schonmann, R. H. and Sidoravicius, V. (2002). Stretched exponential fixation in stochastic Ising models at zero temperature.Comm. Math. Phys.228495–518.
  • [15] Gravner, J., Holroyd, A. E. and Morris, R. (2012). A sharper threshold for bootstrap percolation in two dimensions.Probab. Theory Related Fields1531–23.
  • [16] Holroyd, A. E. (2003). Sharp metastability threshold for two-dimensional bootstrap percolation.Probab. Theory Related Fields125195–224.
  • [17] Holroyd, A. E., Liggett, T. M. and Romik, D. (2004). Integrals, partitions, and cellular automata.Trans. Amer. Math. Soc.3563349–3368.
  • [18] Morris, R. The second order term for bootstrap percolation in two dimensions. Preprint. Available athttp://w3.impa.br/~rob.
  • [19] Morris, R. (2011). Zero-temperature Glauber dynamics on $\mathbb{Z}^{d}$.Probab. Theory Related Fields149417–434.
  • [20] Mountford, T. S. (1995). Critical length for semi-oriented bootstrap percolation.Stochastic Process. Appl.56185–205.
  • [21] Sauer, N. (1972). On the density of families of sets.J. Combin. Theory Ser. A13145–147.
  • [22] Schonmann, R. H. (1990). Critical points of two-dimensional bootstrap percolation-like cellular automata.J. Stat. Phys.581239–1244.
  • [23] Schonmann, R. H. (1992). On the behavior of some cellular automata related to bootstrap percolation.Ann. Probab.20174–193.
  • [24] Shelah, S. (1972). A combinatorial problem; stability and order for models and theories in infinitary languages.Pacific J. Math.41247–261.
  • [25] van Enter, A. C. D. and Fey, A. (2012). Metastability thresholds for anisotropic bootstrap percolation in three dimensions.J. Stat. Phys.14797–112.