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

TL;DR

The paper analyzes the two-dimensional Abelian sandpile model through the lens of site activity, defining $A(R)$ as the total topplings at a site when avalanches are triggered from all sites on a fixed configuration. Using extensive simulations, it shows a finite-size scaling form $P(A,L) \sim L^{-2} F\left(\frac{A}{L^2}\right)$ with $F(u)\sim u^{-1/2}$ for small $u$ and $F(u)\sim \exp(-c_3 u - c_4 u^2)$ for large $u$, and identifies a crossover at $A^*\sim 0.1\,L^2$ between typical and highly excitable sites. The analysis derives consistency conditions for the scaling form, showing $\alpha=\beta=2$ and $\langle A \rangle \sim L^2$, and provides a unified description that links local activity to global avalanche dynamics. Overall, the work offers a locally focused characterization of SOC in sandpile models that complements conventional avalanche statistics and aids interpretations for real systems with partial observations, across system sizes up to $L=160$.

Abstract

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.

Figures (9)

• Figure 1: The heatmap of function $s$ for a configuration of lattice size 200
• Figure 2: The heatmap of function $A$ for the same configuration, normalised by $L^2$
• Figure 3: Plot of the expectation values $\langle A\rangle$ and $\langle s\rangle$ as a function of $L$, verifying that $\langle A\rangle=\langle s\rangle$.
• Figure 4: Log–log plot of $g(a)=LP(A,L)$ versus $a$ for small $a$. The slope of the straight line indicates $g(a) \sim a^{-0.385\pm{0.03}}$. $R^2=0.9839$ for fit. The data can also be fitted fairly well with the form $g(x) \approx c_1 \, x^{-1/2}+c_2$ where $x$ is $a/L^2$ and $c_1,c_2$ are constants numerically determined.
• Figure 5: Log-log plot of $P(A = a)$ vs. $L$ for small fixed values of $a$. The observed slope of $-1$ indicates $P(A = a) \sim 1/L$.