Sandpile Model on Sierpinski Gasket Fractal

TL;DR

This paper studies the sandpile model on the Sierpinski gasket fractal, revealing unique critical behavior with power-law avalanche distributions influenced by the fractal's discrete scale invariance.

Contribution

It introduces the analysis of critical exponents and logarithmic oscillations in avalanche distributions on a fractal lattice, a novel extension of sandpile models.

Findings

Critical exponents for avalanche size, lifetime, and topplings.

Power law distribution with logarithmic oscillations.

Relation of oscillations to discrete scale invariance.

Abstract

We investigate the sandpile model on the two--dimensional Sierpinski gasket fractal. We find that the model displays novel critical behavior, and we analyze the distribution functions of avalanche sizes, lifetimes and topplings and calculate the associated critical exponents $\tau = 1.51 \pm 0.04$, $\alpha = 1.63 \pm 0.04$ and $\mu = 1.36 \pm 0.04$. The avalanche size distribution shows power law behavior modulated by logarithmic oscillations which can be related to the discrete scale invariance of the underlying lattice. Such a distribution can be formally described by introducing a complex scaling exponent ${\tau}^{*} \equiv \tau + i \delta$, where the real part $\tau$ corresponds to the power law and the imaginary part $\delta$ is related to the period of the logarithmic oscillations.