Sandpiles on the Sierpinski gasket

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.