Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Bootstrap Percolation

  • Paolo De Gregorio
  • Aonghus Lawlor
  • Kenneth A. Dawson
Living reference work entry
DOI: https://doi.org/10.1007/978-3-642-27737-5_41-3

Definition of the Subject

In bootstrap percolation, we start with a lattice configuration of occupied and vacant sites. Occupied sites that have less than a certain prescribed number of occupied neighbors are rendered vacant, and as new occupied sites are found to satisfy the same condition, these are also rendered vacant. The process is iterated until eventually no more sites can be removed (if any exist). Bootstrap percolation endeavors to determine whether any occupied sites will survive the culling process and what the macroscopic geometrical properties of the occupied ensembles are. In essence, it is a generalization of conventional percolation which has led to many fruitful insights. A complementary view, which emphasizes the dynamical aspect of the bootstrap process, treats the vacant sites as invasive units and occupied sites as inert ones. Inert sites that are surrounded by too many invasive sites will become irreversibly infected and will start themselves to invade others. At...

Keywords

System Size Vacant Site Regular Lattice Cayley Tree Occupied Site 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Bibliography

  1. Adler J (1991) Bootstrap percolation. Phys A 171(3):453–470CrossRefGoogle Scholar
  2. Adler J, Aharony A (1988) Diffusion percolation. I. Infinite time limit and bootstrap percolation. J Phys A Math Gen 21(6):1387–1404ADSMathSciNetCrossRefGoogle Scholar
  3. Adler J, Lev U (2003) Bootstrap percolation: visualisations and applications. Braz J Phys 33:641ADSCrossRefGoogle Scholar
  4. Adler J, Stauffer D (1990) Evidence for non-universal exponents in bootstrap percolation. J Phys A Math Gen 23:L1119ADSCrossRefGoogle Scholar
  5. Adler J, Palmer RG, Meyer H (1987) Transmission of order in some unusual dilute systems. Phys Rev Lett 58(9):882–885ADSMathSciNetCrossRefGoogle Scholar
  6. Adler J, Stauffer D, Aharony A (1989) Comparison of bootstrap percolation models. J Phys A Math Gen 22:L297ADSCrossRefGoogle Scholar
  7. Aizenman M, Lebowitz JL (1988) Metastability effects in bootstrap percolation. J Phys A Math Gen 21:3801–3813ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. Alvarez-Hamelin JI, Dall’Asta L, Barrat A, Vespignani A (2005) k-Core decomposition: a tool for the analysis of large scale internet graphs. eprint arXiv:cs/0504107Google Scholar
  9. Balogh J, Bollobás B (2003) Sharp thresholds in bootstrap percolation. Phys A Stat Mech Appl 326(3–4):305–312CrossRefzbMATHGoogle Scholar
  10. Balogh J, Bollobás B (2006) Bootstrap percolation on the hypercube. Probab Theory Relat Fields 134(4):624–648MathSciNetCrossRefzbMATHGoogle Scholar
  11. Balogh J, Pete G (1998) Random disease on the square grid. Random Struct Algoritm 13(3–4):409–422MathSciNetCrossRefzbMATHGoogle Scholar
  12. Balogh J, Pittel BG (2007) Bootstrap percolation on the random regular graph. Random Struct Algoritm 30(1–2):257–286MathSciNetCrossRefzbMATHGoogle Scholar
  13. Balogh J, Bollobás B, Morris R (2007) Majority bootstrap percolation on the hypercube. Arxiv preprint mathCO/0702373Google Scholar
  14. Balogh J, Bollobás B, Duminil-Copin H, Morris R (2012) The sharp threshold for bootstrap percolation in all dimensions. Trans Am Math Soc 364(5):2667–2701MathSciNetCrossRefzbMATHGoogle Scholar
  15. Bollobás B (2001) Random graphs. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  16. Branco NS (1993) Probabilistic bootstrap percolation. J Stat Phys 70:1035–1044ADSCrossRefzbMATHGoogle Scholar
  17. Branco NS, Silva CJ (1999) Universality class for bootstrap percolation with m = 3 on the cubic lattice. Int J Mod Phys C 10:921–930ADSCrossRefGoogle Scholar
  18. Branco N, Dos Santos R, de Queiroz S (1984) Bootstrap percolation: a renormalization group approach. J Phys C 17:1909–1921CrossRefGoogle Scholar
  19. Bringmann K, Mahlburg K (2012) Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation. Trans Am Math Soc 364(7):3829–3859MathSciNetCrossRefzbMATHGoogle Scholar
  20. Cerf R, Cirillo E (1999) Finite size scaling in three-dimensional bootstrap percolation. Ann Probab 27(4):1837–1850MathSciNetCrossRefzbMATHGoogle Scholar
  21. Cerf R, Manzo F (2002) The threshold regime of finite volume bootstrap percolation. Stoch Process Appl 101:69–82MathSciNetCrossRefzbMATHGoogle Scholar
  22. Chalupa J, Leath PL, Reich GR (1979) Bootstrap percolation on a Bethe lattice. J Phys C Solid State Phys 12(1):L31–L35ADSCrossRefGoogle Scholar
  23. Chaves C, Koiller B (1995) Universality, thresholds and critical exponents in correlated percolation. Phys A 218(3):271–278CrossRefGoogle Scholar
  24. De Gregorio P, Lawlor A, Bradley P, Dawson KA (2004) Clarification of the bootstrap percolation paradox. Phys Rev Lett 93:025501ADSCrossRefGoogle Scholar
  25. De Gregorio P, Lawlor A, Bradley P, Dawson KA (2005) Exact solution of a jamming transition: closed equations for a bootstrap percolation problem. Proc Natl Acad Sci 102:5669–5673ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. De Gregorio P, Lawlor A, Dawson KA (2006) New approach to study mobility in the vicinity of dynamical arrest; exact application to a kinetically constrained model. Europhys Lett 74:287–293ADSCrossRefGoogle Scholar
  27. Dorogovtsev SN, Goltsev AV, Mendes JFF (2006) k-Core organization of complex networks. Phys Rev Lett 96:040601ADSCrossRefzbMATHGoogle Scholar
  28. Duarte J (1989) Simulation of a cellular automat with an oriented bootstrap rule. Phys A 157(3):1075–1079CrossRefGoogle Scholar
  29. Ertel W, Frobröse K, Jäckle J (1988) Constrained diffusion dynamics in the hard-square lattice gas at high density. J Chem Phys 88:5027–5034ADSCrossRefGoogle Scholar
  30. Fernholz D, Ramachandran V (2003) The giant k-core of a random graph with a specified degree sequence. manuscript, UT-AustinGoogle Scholar
  31. Fredrickson GH, Andersen HC (1984) Kinetic Ising model of the glass transition. Phys Rev Lett 53(13):1244–1247ADSCrossRefGoogle Scholar
  32. Goltsev AV, Dorogovtsev SN, Mendes JFF (2006) k-Core (bootstrap) percolation on complex networks: critical phenomena and nonlocal effects. Phys Rev E 73:056101ADSMathSciNetCrossRefGoogle Scholar
  33. Gravner J, Griffeath D (1996) First passage times for threshold growth dynamics on Z2. Ann Probab 24(4):1752–1778MathSciNetCrossRefzbMATHGoogle Scholar
  34. Gravner J, Holroyd A (2008) Slow convergence in bootstrap percolation. Ann Probab 18(3):909–928MathSciNetCrossRefzbMATHGoogle Scholar
  35. Gravner J, Holroyd A (2009) Local bootstrap percolation. Electron J Probab 14(14):385–399MathSciNetCrossRefzbMATHGoogle Scholar
  36. Gravner J, McDonald E (1997) Bootstrap percolation in a polluted environment. J Stat Phys 87(3):915–927ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. Harris A, Schwarz J (2005) 1/d expansion for k-core percolation. Phys Rev E 72(4):46123ADSCrossRefGoogle Scholar
  38. Holroyd A (2003) Sharp metastability threshold for two-dimensional bootstrap percolation. Probab Theory Relat Field 125(2):195–224MathSciNetCrossRefzbMATHGoogle Scholar
  39. Holroyd A (2006) The metastability threshold for modified bootstrap percolation in d dimensions. Electron J Probab 11:418–433MathSciNetCrossRefzbMATHGoogle Scholar
  40. Holroyd A, Liggett T, Romik D (2004) Integrals, partitions and cellular automata. Trans Am Math Soc 356:3349–3368MathSciNetCrossRefzbMATHGoogle Scholar
  41. Ising E (1925) Beitrag zur Theorie des Ferromagnetismus. Z Phys 31:253–258ADSCrossRefGoogle Scholar
  42. Jäckle J, Krönig A (1994) A kinetic lattice-gas model for the triangular lattice with strong dynamic correlations: I. cself-diffusion. J Phys Condens Matter 6:7633–7653CrossRefGoogle Scholar
  43. Jeng M, Schwarz JM (2007) Comment on “jamming percolation and glass transitions in lattice models”. Phys Rev Lett 98(12):129601ADSCrossRefGoogle Scholar
  44. Kirkpatrick S, Wilcke W, Garner R, Huels H (2002) Percolation in dense storage arrays. Phys A Stat Mech Appl 314(1–4):220–229MathSciNetCrossRefzbMATHGoogle Scholar
  45. Kob W, Andersen HC (1993) Kinetic lattice-gas model of cage effects in high-density liquids and a test of mode-coupling theory of the ideal-glass transition. Phys Rev E 48:4364–4377ADSCrossRefGoogle Scholar
  46. Kogut PM, Leath PL (1981) Bootstrap percolation transitions on real lattices. J Phys C 14(22):3187–3194ADSCrossRefGoogle Scholar
  47. Kurtsiefer D (2003) Threshold value of three dimensional bootstrap percolation. Int J Mod Phys C 14:529ADSCrossRefGoogle Scholar
  48. Lawlor A, De Gregorio P, Bradley P, Sellitto M, Dawson KA (2005) Geometry of dynamically available empty space is the key to near-arrest dynamics. Phys Rev E 72:021401ADSCrossRefGoogle Scholar
  49. Lawlor A, De Gregorio P, Cellai D, Dawson KA (2008) (to be published)Google Scholar
  50. Manna SS (1998) Abelian cascade dynamics in bootstrap percolation. Phys A Stat Theory Phys 261:351–358CrossRefGoogle Scholar
  51. Medeiros M, Chaves C (1997) Universality in bootstrap and diffusion percolation. Phys A 234(3):604–610CrossRefGoogle Scholar
  52. Moukarzel C, Duxbury PM, Leath PL (1997) Infinite-cluster geometry in central-force networks. Phys Rev Lett 78(8):1480–1483ADSCrossRefGoogle Scholar
  53. Mountford TS (1995) Critical length for semi-oriented bootstrap percolation. Stoch Process Appl 56:185–205MathSciNetCrossRefzbMATHGoogle Scholar
  54. Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167ADSMathSciNetCrossRefzbMATHGoogle Scholar
  55. O’Hern C, Silbert L, Liu A, Nagel S (2003) Jamming at zero temperature and zero applied stress: the epitome of disorder. Phys Rev E 68(1):11306CrossRefGoogle Scholar
  56. Onsager L (1944) Crystal statistics. I. A two-dimensional model with an order–disorder transition. Phys Rev 65(3–4):117–149ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. Parisi G, Rizzo T (2006) On k-core percolation in four dimensions. Arxiv preprint cond-mat/0609777Google Scholar
  58. Pittel B, Spencer J, Wormald N (1996) Sudden emergence of a giant k-core in a random graph. J Comb Theory B 67:111–151MathSciNetCrossRefzbMATHGoogle Scholar
  59. Pollak M, Riess I (1975) Application of percolation theory to 2D-3D Heisenberg ferromagnets. Phys Stat Solidi (B) 69(1):K15–K18ADSCrossRefGoogle Scholar
  60. Ritort F, Sollich P (2003) Glassy dynamics of kinetically constraint models. Adv Phys 52:219–342ADSCrossRefGoogle Scholar
  61. Sabhapandit S, Dhar D, Shukla P (2002) Hysteresis in the random-field Ising model and bootstrap percolation. Phys Rev Lett 88(19):197202ADSCrossRefGoogle Scholar
  62. Sahimi M (1994) Applications of percolation theory. Taylor/Francis, LondonGoogle Scholar
  63. Schonmann R (1990a) Critical points of two-dimensional bootstrap percolation-like cellular automata. J Stat Phys 58(5):1239–1244ADSMathSciNetCrossRefzbMATHGoogle Scholar
  64. Schonmann R (1990b) Finite size scaling behavior of a biased majority rule cellular automaton. Phys A 167(3):619–627MathSciNetCrossRefGoogle Scholar
  65. Schonmann R (1992) On the behavior of some cellular automata related to bootstrap percolation. Ann Probab 20(1):174–193MathSciNetCrossRefzbMATHGoogle Scholar
  66. Schwarz JM, Liu AJ, Chayes LQ (2006) The onset of jamming as the sudden emergence of an infinite k-core cluster. Europhys Lett (EPL) 73(4):560–566ADSCrossRefGoogle Scholar
  67. Sellitto M, Biroli G, Toninelli C (2005) Facilitated spin models on Bethe lattice: bootstrap percolation, mode-coupling transition and glassy dynamics. Europhys Lett 69(4):496–502ADSCrossRefGoogle Scholar
  68. Smirnov S, Werner W (2001) Critical exponents for 2D percolation. Math Res Lett 8:729–744MathSciNetCrossRefzbMATHGoogle Scholar
  69. Stauffer D, Aharony A (1992) Introduction to percolation theory. Taylor/Francis, LondonzbMATHGoogle Scholar
  70. Toninelli C, Biroli G, Fisher DS (2006) Jamming percolation and glass transitions in lattice models. Phys Rev Lett 96(3):035702ADSCrossRefGoogle Scholar
  71. Toninelli C, Biroli G, Fisher D (2007) Toninelli, Biroli, and Fisher reply. Phys Rev Lett 98(12):129602ADSCrossRefGoogle Scholar
  72. Treaster M, Conner W, Gupta I, Nahrstedt K (2006) Contagalert: using contagion theory for adaptive, distributed alert propagation. In: Network computing and applications, NCA 2006, pp 126–136Google Scholar
  73. van Enter A (1987) Proof of Straley’s argument for bootstrap percolation. J Stat Phys 48(3):943–945ADSCrossRefzbMATHGoogle Scholar
  74. van Enter A, Hulshof T (2007) Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections. J Stat Phys 128(6):1383–1389ADSMathSciNetCrossRefzbMATHGoogle Scholar
  75. van Enter ACD, Adler J, Duarte JAMS (1990) Finite size effects for some bootstrap percolation models. J Stat Phys 60:323–332ADSMathSciNetCrossRefGoogle Scholar
  76. van Enter A, Adler J, Duarte J (1991) Finite-size effects for some bootstrap percolation models, addendum. J Stat Phys 62:505–506ADSCrossRefGoogle Scholar
  77. Widom B (1974) The critical point and scaling theory. Physica 73(1):107–118ADSCrossRefGoogle Scholar
  78. Wilson K (1983) The renormalization group and critical phenomena. Rev Mod Phys 55:583–600ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Paolo De Gregorio
    • 1
  • Aonghus Lawlor
    • 2
  • Kenneth A. Dawson
    • 2
  1. 1.Dipartimento di Fisica e Astronomia, Università di PadovaINFN. Sezione di PadovaPadovaItaly
  2. 2.School of Chemistry and Chemical BiologyUniversity College DublinDublinIreland