Sandpiles on the Sierpinski gasket

TL;DR

This paper investigates the sandpile model on the Sierpinski gasket, deriving exact critical exponents for wave and avalanche distributions through simulations and extended theory, confirming predictions with numerical data.

Contribution

It extends the theory of waves in sandpile models to fractal structures and derives exact critical exponents for the Sierpinski gasket case.

Findings

Exact value for wave size exponent: τ_w = ln(9/5)/ln(3)

Conjectured avalanche size exponent: τ = 1 + τ_w = ln(27/5)/ln(3)

Numerical results agree with theoretical predictions.

Abstract

We perform extensive simulations of the sandpile model on a Sierpinski gasket. Critical exponents for waves and avalanches are determined. We extend the existing theory of waves to the present case. This leads to an exact value for the exponent $\tau_w$ which describes the distribution of wave sizes: $\tau_w = \ln{(9/5)}/\ln{3}$. Numerically, it is found that the number of waves in an avalanche is proportional to the number of distinct sites toppled in the avalanche. This leads to a conjecture for the exponent $\tau$ that determines the distribution of avalanche sizes: $\tau=1+\tau_w = \ln{(27/5)}/\ln{3}$. Our predictions are in good agreement with the numerical results.