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Geometry and correlations in the BTW sandpile model

The Abelian sandpile model is one of the simplest realizations of self-organized criticality, where scale-invariant behaviour emerges without fine tuning of parameters. Traditionally, the system is characterized using avalanche observables such as size and duration. However, this description is local in nature and does not fully capture the internal structure of the critical state. In this work, I study the activity field, which measures how often a given site participates in avalanches triggered elsewhere. This provides a global description of the system and reveals nontrivial spatial organization of the critical attractor. Recent studies suggest that activity exhibits scaling behaviour and is closely related to the geometry of avalanche clusters. The main goal is to understand whether the activity field is multifractal and how spatial correlations emerge in the steady state. I investigate scaling of moments of activity, two-point correlations, and the relation between activity and excitability fields. A key question is whether these properties are universal or depend on the driving protocol.

Fig: Time evolution of the BTW sandpile model showing avalanche activity starting from empty configuration to adding 50k sand grains at center.

This approach provides a unified picture connecting avalanche statistics, spatial structure, and response functions, and may offer a new way to characterize critical states in self-organized systems.

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