Avalanche size exponent of the two-dimensional Abelian sandpile

For the Bak-Tang-Wiesenfeld Abelian sandpile on the L x L square lattice with open boundaries, determine whether the avalanche size distribution obeys finite-size scaling $P(s) = s^{-\tau} F(s/L^D)$, and if so the exact $\tau$ and $D$ (proposed values of $\tau$ lie near 1.2 to 1.3); if not, characterize the multifractal spectrum of avalanche sizes. An answer is exact exponents or a proof of multifractal scaling.

An exact computation of avalanche statistics using the Abelian structure, or a proof that simple finite-size scaling fails, consistent with simulations.