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Self-Organized Criticality in Glassy Spin Systems Requires a Diverging Number of Neighbors

We investigate the conditions required for general spin systems with frustration and disorder to display self-organized criticality, a property which so far has been established only for the fully-connected infinite-range Sherrington-Kirkpatrick Ising spin-glass model [Phys. Rev. Lett. 83, 1034 (1999)]. Here we study both avalanche and magnetization jump distributions triggered by an external magnetic field, as well as internal field distributions in the short-range Edwards-Anderson Ising spin glass for various space dimensions between 2 and 8, as well as the fixed-connectivity mean-field Viana-Bray model. Our numerical results, obtained on systems of unprecedented size, demonstrate that self-organized criticality is recovered only in the strict limit of a diverging number of neighbors, and is not a generic property of spin-glass models in finite space dimensions.