Abstract.
We study a model of ‘‘organized’’ criticality, where a single avalanche propagates through an a priori static (i.e., organized) sandpile configuration. The latter is chosen according to an i.i.d. distribution from a Borel probability measure ρ 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 ρ 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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Mathematics Subject Classification (2000): 60K35, 82C20, 82C44
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Biskup, M., Blanchard, P., Chayes, L. et al. Phase transition and critical behavior in a model of organized criticality. Probab. Theory Relat. Fields 128, 1–41 (2004). https://doi.org/10.1007/s00440-003-0269-z
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DOI: https://doi.org/10.1007/s00440-003-0269-z
Keywords
- Phase Transition
- Probability Measure
- Critical Exponent
- Rooted Tree
- Close Connection