Abstract
In this note we analyse an anisotropic, two-dimensional bootstrap percolation model introduced by Gravner and Griffeath. We present upper and lower bounds on the finite-size effects. We discuss the similarities with the semi-oriented model introduced by Duarte.
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Adler, J.: Bootstrap percolation. Physica A 171, 452–470 (1991)
Adler, J., Duarte, J.A.M.S., van Enter, A.C.D.: Finite-size effects for some bootstrap percolation models. J. Stat. Phys. 60, 323–332 (1990)
Adler, J., Duarte, J.A.M.S., van Enter, A.C.D.: Finite-size effects for some bootstrap percolation models, addendum. J. Stat. Phys. 62, 505–506 (1991)
Adler, J., Lev, U.: Bootstrap percolation: visualizations and applications. Braz. J. Phys. 33, 641–644 (2003)
Aizenman, M., Lebowitz, J.L.: Metastability effects in bootstrap percolation. J. Phys. A: Math. Gen. 21, 3801–3813 (1988)
Balogh, J., Bollobas, B.: Sharp thresholds in bootstrap percolation. Physica A 326, 305–312 (2003)
Biroli, G., Fisher, D.S., Toninelli, C.: Jamming percolation and glass transitions in lattice models. Phys. Rev. Lett. 96, 035702 (2006)
Biroli, G., Fisher, D.S., Toninelli, C.: Cooperative behavior of kinetically constrained lattice gas models of glassy dynamics. J. Stat. Phys. 120, 167–238 (2005)
Cerf, R., Cirillo, E.M.N.: Finite size scaling in three-dimensional bootstrap percolation. Ann. Probab. 27, 1833–1850 (1999)
Connelly, R., Rybnikov, K., Volkov, S.: Percolation of the loss of tension in an infinite triangular lattice. J. Stat. Phys. 105, 143–171 (2001)
De Gregorio, P., Lawlor, A., Bradley, P., Dawson, K.A.: Cellular automata with rare events; Resolution of an outstanding problem in the bootstrap percolation model. In: Cellular Automata. Amsterdam Proceedings, Lecture Notes in Computer Science, vol. 3305, pp. 365–374. Springer, Berlin (2004)
De Gregorio, P., Lawlor, A., Bradley, P., Dawson, K.A.: Clarification of the bootstrap percolation paradox. Phys. Rev. Lett. 93, 025501 (2004)
Duarte, J.A.M.S.: Simulation of a cellular automaton with an oriented bootstrap rule. Physica A 157, 1075–1079 (1989)
Gravner, J., Griffeath, D.: First passage times for threshold growth dynamics on ℤ2. Ann. Probab. 24, 1752–1778 (1996) (see also Griffeath’s webpage: http://psoup.math.wisc.edu/kitchen.html)
Gravner, J., Griffeath, D.: Scaling laws for a class of cellular automaton growth rules. In: Proceedings 1998 Erdös Center Workshop on Random Walks, pp. 167–186 (1999)
Gravner, J., Holroyd, A.: Slow convergence in bootstrap percolation. arXiv: 0705.1347 (2007) (see also Holroyd’s webpage: http://www.math.ubc.ca/holroyd/)
Holroyd, A.: Sharp metastability threshold for two-dimensional bootstrap percolation. Probab. Theory Relat. Fields 125, 195–224 (2003)
Holroyd, A.: The metastability threshold for modified bootstrap percolation in d dimensions. Electron. J. Probab. 11, 418–433 (2006)
Hulshof, W.J.T.: The similarities between an unbalanced and an oriented bootstrap percolation model. Groningen bachelor thesis (2007)
Kozma, R., Puljic, M., Balister, P., Bollobas, B., Freeman, W.J.: Phase transitions in the neuropercolation model of neural populations with mixed local and nonlocal interactions. Biol. Cybern. 92, 367–379 (2005)
Kirkpatrick, S., Wilcke, W.W., Garner, R.B., Huels, H.: Percolation in dense storage arrays. Physica A 314, 220–229 (2002)
Lee, I.H., Valentiniy, A.: Noisy contagion without mutation. Rev. Econ. Stud. 67, 47–56 (2000)
Lenormand, R.: Pattern growth and fluid displacement through porous media. Physica A 140, 114–123 (1986)
Mountford, T.S.: Critical lengths for semi-oriented bootstrap percolation. Stoch. Proc. Appl. 95, 185–205 (1995)
Mountford, T.S.: Comparison of semi-oriented bootstrap percolation models with modified bootstrap percolation. In: Boccara, N., Goles, E., Martinez, S. (eds.) Cellular Automata and Cooperative Systems. NATO ASI Proceedings, pp. 519–525. Kluwer Acadamic, Dordrecht (1993)
Ritort, F., Sollich, P.: Glassy dynamics of constrained models. Adv. Phys. 52, 219–342 (2003)
Sabhapandit, S., Dhar, D., Shukla, P.: Hysteresis in the random-field Ising model and bootstrap percolation. Phys. Rev. Lett. 88, 197202 (2002)
Schonmann, R.H.: Critical points of 2-dimensional bootstrap percolation-like cellular automata. J. Stat. Phys. 58, 1239–1244 (1990)
Schonmann, R.H.: On the behavior of some cellular automata related to bootstrap percolation. Ann. Probab. 20, 174–193 (1992)
Treaster, M., Conner, W., Gupta, I., Nahrstedt, K.: ContagAlert: using contagion theory for adaptive, distributed alert propagation. In: Fifth IEEE International Symposium on Network Computing and Applications, pp. 126–136 (2006)
van Enter, A.C.D.: Proof of Straley’s argument for bootstrap percolation. J. Stat. Phys. 48, 943–945 (1987)
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van Enter, A.C.D., Hulshof, T. Finite-Size Effects for Anisotropic Bootstrap Percolation: Logarithmic Corrections. J Stat Phys 128, 1383–1389 (2007). https://doi.org/10.1007/s10955-007-9377-y
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DOI: https://doi.org/10.1007/s10955-007-9377-y