Electronic Journal of Probability

Local Bootstrap Percolation

Janko Gravner and Alexander Holroyd

Full-text: Open access


We study a variant of bootstrap percolation in which growth is restricted to a single active cluster. Initially there is a single active site at the origin, while other sites of $\mathbb{Z}^2$ are independently occupied with small probability $p$, otherwise empty. Subsequently, an empty site becomes active by contact with two or more active neighbors, and an occupied site becomes active if it has an active site within distance 2. We prove that the entire lattice becomes active with probability $\exp [\alpha(p)/p]$, where $\alpha(p)$ is between $-\pi^2/9+c\sqrt p$ and $-\pi^2/9+C\sqrt p(\log p^{-1})^3$. This corrects previous numerical predictions for the scaling of the correction term.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 14, 385-399.

Accepted: 9 February 2009
First available in Project Euclid: 1 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

bootstrap percolation cellular automaton crossover finite-size scaling metastability

This work is licensed under aCreative Commons Attribution 3.0 License.


Gravner, Janko; Holroyd, Alexander. Local Bootstrap Percolation. Electron. J. Probab. 14 (2009), paper no. 14, 385--399. doi:10.1214/EJP.v14-607. https://projecteuclid.org/euclid.ejp/1464819475

Export citation


  • J. Adler and U. Lev. Bootstrap percolation: Visualizations and applications. Brazilian J. Phys., 33(3):641–644, 2003.
  • J. Adler, D. Stauffer, and A. Aharony. Comparison of bootstrap percolation models. J. Phys. A, 22:L297–L301, 1989.
  • M. Aizenman and J. L. Lebowitz. Metastability effects in bootstrap percolation. J. Phys. A, 21(19):3801–3813, 1988.
  • J. Balogh and B. Bollobas. Sharp thresholds in bootstrap percolation. Physica A, 326(3):305–312, 2003.
  • N. Cancrini, F. Martinelli, C. Roberto, and C. Toninelli. Kinetically constrained spin models. Probab. Theory Related Fields 140(3-4): 459–504, 2008.
  • P. De Gregorio, A. Lawlor, P. Bradley, and K. A. Dawson. Exact solution of a jamming transition: closed equations for a bootstrap percolation problem. Proc. Natl. Acad. Sci. USA, 102(16):5669–5673 (electronic), 2005.
  • P. De Gregorio, A. Lawlor, and K. A. Dawson. New approach to study mobility in the vicinity of dynamical arrest; exact application to a kinetically constrained model. Europhys. Lett., 74(2):287–293, 2006.
  • L. R. Fontes, R. H. Schonmann, and V. Sidoravicius. Stretched exponential fixation in stochastic Ising models at zero temperature. Comm. Math. Phys., 228(3):495–518, 2002.
  • K. Frobose. Finite-size effects in a cellular automaton for diffusion. J. Statist. Phys., 55(5-6):1285–1292, 1989.
  • J. Gravner, and A. E. Holroyd. Slow convergence in bootstrap percolation. Ann. Appl. Prob., 18(3):909–928, 2008.
  • G. R. Grimmett. Percolation. Springer-Verlag, second edition, 1999.
  • A. E. Holroyd. Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Related Fields, 125(2):195–224, 2003.
  • A. E. Holroyd. The metastability threshold for modified bootstrap percolation in d dimensions. Electron. J. Probab., 11:no. 17, 418–433 (electronic), 2006.
  • A. E. Holroyd, T. M. Liggett, and D. Romik. Integrals, partitions, and cellular automata. Trans. Amer. Math. Soc., 356(8):3349–3368, 2004.
  • R. H. Schonmann, On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab., 20(1): 174–193, 1992.
  • D. Stauffer, Work described in Adler and Lev, 2003.
  • A. C. D. van Enter. Proof of Straley's argument for bootstrap percolation. J. Statist. Phys., 48(3-4):943–945, 1987.