Scaling Properties of Avalanche Activity in the Two-Dimensional Abelian Sandpile Model

We study the scaling properties of avalanche activity in the two-dimensional Abelian sandpile model. Instead of the conventional avalanche size distribution, we analyze the site activity distribution, which measures how often a site participates in avalanches when grains are added across the lattice. Using numerical simulations for system sizes up to $L = 160$, averaged over $10^4$ configurations, we determine the probability distribution $P(A, L)$ of site activities. The results show that $P(A, L)$ follows a finite-size scaling form $$P(A, L) \sim L^{-2} F\Big(\frac{A}{L^2}\Big).$$ For small values $A \ll L^2$ the scaling function behaves as $$F(u) \sim u^{-1/2}, \quad \text{corresponding to} \quad P(A) \sim \frac{1}{L},$$ while for large activities $A \sim O(L^2)$ the distribution decays as $$F(u) \sim \exp\big(-c_3 u - c_4 u^2\big).$$ The crossover between these two regimes occurs at $$A^* \sim 0.1 \, L^2,$$ marking the threshold between typical and highly excitable sites. This characterization of local avalanche activity provides complementary information to the usual avalanche size statistics, highlighting how local regions serve as frequent conduits for critical dynamics. These results may help connect sandpile models to real-world self-organized critical systems where only partial local activity can be observed.