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Maximal avalanches in the Bak-Sneppen model

  • Alexis Gillett (a1), Ronald Meester (a1) and Peter van der Wal (a2)
Abstract

We study the durations of the avalanches in the maximal avalanche decomposition of the Bak-Sneppen evolution model. We show that all the avalanches in this maximal decomposition have infinite expectation, but only ‘barely’, in the sense that if we made the appropriate threshold a tiny amount smaller (in a certain sense), then the avalanches would have finite expectation. The first of these results is somewhat surprising, since simulations suggest finite expectations.

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Copyright
Corresponding author
Postal address: Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands.
∗∗ Email address: ajg@cs.vu.nl
∗∗∗ Email address: rmeester@cs.vu.nl
∗∗∗∗ Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: vanderwal@eurandom.tue.nl
References
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[1] Bak, P. (1996).” How Nature Works. ” Copernicus, New York.
[2] Bak, P. and Paczuski, M. (1995). “Complexity, contingency and criticality.” Proc. Nat. Acad. Sci. USA 92, 66896696.
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[4] Bak, P., Tang, C. and Wiesenfeld, K. (1987). “Self-organized criticality: an explanation of 1/f noise.” Phys. Rev. Lett. 59, 381384.
[5] Feller, W. (1966). “An Introduction to Probability Theory and Its Applications, Vol. 2.” John Wiley, New York.
[6] Gillett, A., Meester, R. and Nuyens, M. (2006). Bounds for avalanche critical values of the Bak–Sneppen model. To appear in Markov Process. Relat. Fields.
[7] Lee, C., Zhu, X. and Gao, K. (2003). “Avalanche dynamics in the {Bak–Sneppen} evolution model observed with a standard distribution width of fitness.” Nonlinearity 16, 2533.
[8] Li, W. and Cai, X. (2000). “Analytic results for scaling function and moments for a different type of avalanche in the {Bak–Sneppen} evolution model.” Phys. Rev. E 62, 77437747.
[9] Maslov, S. (1996). “Infinite series of exact equations in the {Bak–Sneppen} model of biological evolution.” Phys. Rev. Lett. 77, 11821185.
[10] Meester, R. and Znamenski, D. (2003). “Limit behavior of the {Bak–Sneppen} evolution model.” Ann. Prob. 31, 19861986.
[11] Meester, R. and Znamenski, D. (2004). “Critical thresholds and the limit distribution in the {Bak–Sneppen} model.” Commun. Math. Phys. 246, 6386.
[12] Paczuski, S., Maslov, M. and Bak, P. (1996). “Avalanche dynamics in evolution, growth, and depinning models.” Phys. Rev. E 53, 414443.
[13] Tabelow, K. (2001). “Gap function in the finite {Bak–Sneppen} model.” Phys. Rev. E 63, 047101.
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Journal of Applied Probability
  • ISSN: 0021-9002
  • EISSN: 1475-6072
  • URL: /core/journals/journal-of-applied-probability
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