Skip to main content Accessibility help
×
Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-15T10:21:49.330Z Has data issue: false hasContentIssue false

7 - Monotone cellular automata

Published online by Cambridge University Press:  21 July 2017

Robert Morris
Affiliation:
Estrada Dona Castorina
Anders Claesson
Affiliation:
University of Iceland, Reykjavik
Mark Dukes
Affiliation:
University College Dublin
Sergey Kitaev
Affiliation:
University of Strathclyde
David Manlove
Affiliation:
University of Glasgow
Kitty Meeks
Affiliation:
University of Glasgow
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] M., Aizenman and J.L., Lebowitz, Metastability effects in bootstrap percolation, J. Phys. A., 21 (1988), 3801–3813.Google Scholar
[2] R., Arratia, Site recurrence for annihilating random walks on Zd, Ann. Prob., 11 (1983), 706–713.Google Scholar
[3] P., Bak, C., Tang and K., Wiesenfeld, Self-organized criticality: an explanation of 1/f noise, Phys. Rev. Letters, 59 (1987), 381–384.Google Scholar
[4] P., Balister, B., Bollobás, M.J., Przykucki and P.J., Smith, Subcritical U-bootstrap percolation models have non-trivial phase transitions, Trans. Amer. Math. Soc., 368 (2016), 7385–7411.Google Scholar
[5] J., Balogh and B., Bollobás Bootstrap percolation on the hypercube, Prob. Theory Rel. Fields, 134 (2006), 624–648.Google Scholar
[6] J., Balogh, B., Bollobás H., Duminil-Copin and R., Morris, The sharp threshold for bootstrap percolation in all dimensions, Trans. Amer. Math. Soc., 364 (2012), 2667–2701.Google Scholar
[7] J., Balogh, B., Bollobás and R., Morris, Majority bootstrap percolation on the hypercube, Combin. Probab. Computing, 18 (2009), 17–51.Google Scholar
[8] J., Balogh, B., Bollobás and R., Morris, Bootstrap percolation in three dimensions, Ann. Prob., 37 (2009), 1329–1380.Google Scholar
[9] J., Balogh, B., Bollobás and R., Morris, Bootstrap percolation in high dimensions, Combin. Probab. Computing, 19 (2010), 643–692.Google Scholar
[10] J., Balogh, Y., Peres and G., Pete, Bootstrap percolation on infinite trees and non-amenable groups, Combin. Prob. Computing, 15 (2006), 715–730.Google Scholar
[11] J., Balogh and B., Pittel, Bootstrap percolation on the random regular graph, Random Structures Algorithms, 30 (2007), 257–286.Google Scholar
[12] J., van den Berg and H., Kesten, Inequalities with applications to percolation and reliability, J. Appl. Prob., 22 (1985), 556–569.Google Scholar
[13] B., Bollobás H., Duminil-Copin, R., Morris and P., Smith, Universality of two-dimensional critical cellular automata, Proc. London Math. Soc., to appear, arXiv:1406.6680.
[14] B., Bollobás H., Duminil-Copin, R., Morris and P., Smith, The sharp threshold for the Duarte model, Ann. Prob., to appear.
[15] B., Bollobás K., Gunderson, C., Holmgren, S., Janson and M., Przykucki, Bootstrap percolation on Galton-Watson trees, Electron. J. Prob., 19 (2014), 1–27.Google Scholar
[16] B., Bollobás S., Griffiths, R., Morris, L., Rolla and P., Smith, Nucleation and growth in two dimensions, submitted, arXiv:1508.06267.
[17] B., Bollobás P.J., Smith and A.J., Uzzell, Monotone cellular automata in a random environment, Combin. Probab. Computing, 24 (2015), 687–722.Google Scholar
[18] N., Cancrini, F., Martinelli, C., Roberto and C., Toninelli, Kinetically constrained spin models, Prob. Theory Rel. Fields, 140 (2008), 459– 504.Google Scholar
[19] R., Cerf and E.N.M., Cirillo, Finite size scaling in three-dimensional bootstrap percolation, Ann. Prob., 27 (1999), 1837–1850.Google Scholar
[20] R., Cerf and F., Manzo, The threshold regime of finite volume bootstrap percolation, Stochastic Proc. Appl., 101 (2002), 69–82.Google Scholar
[21] R., Cerf and F., Manzo, A d-dimensional nucleation and growth model, Prob. Theory Rel. Fields, 155 (2013), 427–449.Google Scholar
[22] R., Cerf and F., Manzo, Nucleation and growth for the Ising model in d dimensions at very low temperatures, Ann. Prob., 41 (2013), 3697–3785.Google Scholar
[23] J., Chalupa, P.L., Leath and G.R., Reich, Bootstrap percolation on a Bethe latice, J. Phys. C., 12 (1979), L31–L35.Google Scholar
[24] P., Dehghanpour and R.H., Schonmann, A nucleation-and-growth model, Prob. Theory Rel. Fields, 107 (1997), 123–135.Google Scholar
[25] P., Dehghanpour and R.H., Schonmann, Metropolis dynamics relaxation via nucleation and growth, Comm. Math. Phys., 188 (1997), 89–119.Google Scholar
[26] A.M.S., Duarte, Simulation of a cellular automaton with an oriented bootstrap rule, Phys. A, 157 (1989), 1075–1079.Google Scholar
[27] H., Duminil-Copin and A., Holroyd, Sharp metastability for threshold growth models, manuscript, available at http://www.unige.ch/~duminil.
[28] H., Duminil-Copin and A.C.D. van, Enter, Finite volume Bootstrap Percolation with balanced threshold rules on Z2, Ann. Prob., 41 (2013), 1218–1242.Google Scholar
[29] A.C.D. van, Enter, Proof of Straley's argument for bootstrap percolation, J. Stat. Phys., 48 (1987),943–945.Google Scholar
[30] A.C.D. van, Enter and A., Fey, Metastability threshold for anisotropic bootstrap percolation in three dimensions, J. Stat. Phys., 147 (2012), 97–112.Google Scholar
[31] A.C.D. van, Enter and W.J.T., Hulshof, Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections, J. Stat. Phys., 128 (2007), 1383–1389.Google Scholar
[32] P., Erdʺos and P., Ney, Some Problems on Random Intervals and Annihilating Particles, Ann. Prob., 2 (1974), 828–839.Google Scholar
[33] A., Fey, R., Meester and F., Redig, Stabilizability and percolation in the infinite volume sandpile model, Ann. Prob., 37 (2009), 654–675.Google Scholar
[34] L.R., Fontes, R.H., Schonmann and V., Sidoravicius, Stretched Exponential Fixation in Stochastic Ising Models at Zero Temperature, Commun. Math. Phys., 228 (2002), 495–518.Google Scholar
[35] J., Gravner and A.E., Holroyd, Slow convergence in bootstrap percolation, Ann. Appl. Prob., 18 (2008), 909–928.Google Scholar
[36] J., Gravner, A.E., Holroyd and R., Morris, A sharper threshold for bootstrap percolation in two dimensions, Prob. Theory Rel. Fields, 153 (2012), 1–23.Google Scholar
[37] J., Gravner and D., Griffeath, Scaling laws for a class of critical cellular automaton growth rules, In: RandomWalks (Budapest, 1998), Bolyai Soc. Math. Stud., 9 (1999), 167–186.Google Scholar
[38] J., Gravner and E., McDonald, Bootstrap percolation in a polluted environment, J. Stat. Phys. 87 (1997), 915–927.Google Scholar
[39] A., Holroyd, Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation, Prob. Theory Rel. Fields, 125 (2003), 195– 224.Google Scholar
[40] A., Holroyd, The metastability threshold for modified bootstrap percolation in d dimensions, Electron. J. Prob., 11 (2006), 418–433.Google Scholar
[41] A.E., Holroyd, T.M., Liggett and D., Romik, Integrals, partitions, and cellular automata, Trans. Amer. Math. Soc., 356 (2004), 3349–3368.Google Scholar
[42] S., Janson, T., Łuczak, T., Turova and T., Vallier, Bootstrap percolation on the random graph G(n, p), Ann. Appl. Prob., 22 (2012), 1989– 2047.Google Scholar
[43] F., Martinelli, Lectures on Glauber dynamics for discrete spin models, Lectures on Probability Theory and Statistics, Springer Lecture Notes in Mathematics, 1717 (1998), 93–191.
[44] F., Martinelli and C., Toninelli, Towards a universality picture for the relaxation to equilibrium of kinetically constrained models, arXiv:1701.00107.
[45] R., Morris, Zero-temperature Glauber dynamics on Zd, Prob. Theory Rel. Fields, 149 (2011), 417–434.Google Scholar
[46] R., Morris, The second term for bootstrap percolation in two dimensions, manuscript, available at http://w3.impa.br/~rob/.
[47] R., Morris, Bootstrap percolation and other automata, European J. Combin., to appear.
[48] N., Morrison and J.A., Noel, Extremal Bounds for Bootstrap Percolation in the Hypercube, arXiv:1506.04686.
[49] T.S., Mountford, Critical length for semi-oriented bootstrap percolation, Stochastic Process. Appl., 56 (1995), 185–205.Google Scholar
[50] S., Nanda, C.M., Newman and D., Stein, Dynamics of Ising spin systems at zero temperature, In On Dobrushin's way (From Probability Theory to Statistical Mechanics), eds. R. Minlos, S. Shlosman and Y. Suhov, Am. Math. Soc. Transl., 198 (2000), 183–194.Google Scholar
[51] J. von, Neumann, Theory of Self-Reproducing Automata. Univ. Illinois Press, Urbana, 1966.
[52] C.M., Newman and D., Stein, Zero-temperature dynamics of Ising spin systems following a deep quench: results and open problems, Physica A, 279 (2000), 159–168.Google Scholar
[53] D., Reimer, Proof of the van den Berg–Kesten Conjecture, Combin. Prob. Computing, 9 (2000), 27–32.Google Scholar
[54] R.H., Schonmann, On the behaviour of some cellular automata related to bootstrap percolation, Ann. Prob., 20 (1992), 174–193.Google Scholar
[55] A., Vespagnani, R., Dickman, M., Muñoz and S., Zapperi, Absorbingstate phase transitions in fixed-energy sandpiles, Phys. Rev. E, 62 (2000), 45–64.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×