We consider a directed Abelian sandpile on a strip of size $2\times n$, driven by adding a grain randomly at the left boundary after every $T$ timesteps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeros in the ternary-base representation of the position of a random walker on a ring of size $3^n$. We find that while the fluctuations of mass have a power spectrum that varies as $1/f$ for frequencies in the range $3^{-2n}\ll f\ll 1/T$, the activity fluctuations in the same frequency range have a power spectrum that is linear in $f$.

• Received 23 March 2012

DOI:https://doi.org/10.1103/PhysRevE.85.061114