Power spectrum of mass and activity fluctuations in a sandpile

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 $T$ time-steps. We establish the exact equivalence of the problem of mass fluctuations in the steady state and the number of zeroes 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 $1\mathrm{/}f$ for frequencies in the range $3^{-2n}\ll f\ll 1\mathrm{/}T$, the activity fluctuations in the same frequency range have a power spectrum that is linear in $f$.