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Many phenomena whose evolution takes place in terms of avalanches are usually
characterized by the statistics of their size, giving rise in some cases to the
so-famous power-law distribution of avalanche sizes. [1]. Less attention has
been paid to the dynamical structure that all these events define in time;
however, recent research has unveiled a complex, self-similar hierarchy of
waiting times, different from the trivial Poisson behavior expected previously
for some of these systems. The distributions of waiting times between these
events verify a scaling law for different event sizes, which can be viewed as
equivalent to the invariance of the system under a renormalization-group
transformation, analogous to that existing in the Ising model in its critical
point [3]. So, the only difference between extreme events and ordinary events is in the scale of observation. Further, this scaling is linked in a fundamental way with the existence of correlations in the process, probably long-range
correlations.
We briefly review results obtained from observational or experimental data on diverse systems, including earthquakes [3], fractures [4], tsunamis [5], extreme climatic events [6], solar flares [7,8], forest fires [9], and financial indices [10]. In the case of the fracture-earthquake system the same scaling law is surprisingly valid from nanofractures to very large earthquakes, covering more than 30 orders of magnitude in the range of energies (sizes) for which it is valid. Curiously, the spatial structure of written language has the same quantitative behavior as the dynamics of some of these systems [11]. References [1] P. Bak. How Nature Works: The Science of Self-Organized Criticality. Copernicus, New York, 1996. [2] A. Corral. Renormalization-Group Transformations and Correlations of Seismicity. Phys. Rev. Lett., 95:028501, 2005. [3] A. Corral. Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes. Phys. Rev. Lett., 92:108501, 2004. [4] J. Davidsen, S. Stanchits and G. Dresen. Scaling and universality in rock fracture. Phys. Rev. Lett., 98:125502, 2007. [5] E. L. Geist and T. Parsons. Distribution of tsunami interevent times. Geophys. Res. Lett., 35:L02612, 2008. [6] A. Bunde, J. F. Eichner, J. W. Kantelhardt and S. Havlin. Long-term memory: a natural mechanism for the clustering of extreme events and anomalous residual times in climate records. Phys. Rev. Lett., 94:048701, 2005. [7] L. de Arcangelis, C. Godano, E. Lippiello, and M. Nicodemi. Universality in Solar Flare and Earthquake Occurrence Phys. Rev. Lett., 96:051102, 2006. [8] M. Baiesi, M. Paczuski and A. L. Stella. Intensity thresholds and the statistics of the temporal occurrence of solar flares. Phys. Rev. Lett., 96:051103, 2006. [9] A. Corral, L. Telesca and R. Lasaponara. Scaling and correlations in the dynamics of forest-fire occurrence. Phys. Rev. E, 77:016101, 2008. [10] K. Yamasaki, L. Muchnik, S. Havlin, A. Bunde and H.E. Stanley. Scaling and memory in volatility return intervals in financial markets. Proc. Natl. Acad. Sci. USA, 102:9424--9428, 2005. [11] A. Corral, R. Ferrer i Cancho and A. Dìaz-Guilera. Complex universal structures in written language. preprint, 2008. |