Phase transition and critical behavior in a model of organized criticality

Details Phase transition and critical behavior in a model of organized criticality Phase transition and critical behavior in a model of organized criticality

2002-06-21

arxiv.org/abs/math/0206232v2

Probab. Theory Rel. Fields 128 (2004), no. 1, 1-41.

Mathematics

Probability

37 pages, version to appear in Prob. Theory Rel. Fields

Scientific paper

10.1007/s00440-003-0269-z

We study a model of ``organized'' criticality, where a single avalanche propagates through an \textit{a priori} static (i.e., organized) sandpile configuration. The latter is chosen according to an i.i.d. distribution from a Borel probability measure $\rho$ on $[0,1]$. The avalanche dynamics is driven by a standard toppling rule, however, we simplify the geometry by placing the problem on a directed, rooted tree. As our main result, we characterize which $\rho$ are critical in the sense that they do not admit an infinite avalanche but exhibit a power-law decay of avalanche sizes. Our analysis reveals close connections to directed site-percolation, both in the characterization of criticality and in the values of the critical exponents.

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