Avalanche size exponent of the two-dimensional Abelian sandpile
In plain words
The Bak-Tang-Wiesenfeld sandpile is the original model of self-organized criticality. Many of its properties are known exactly, yet the power law of its avalanche sizes in two dimensions has not been determined, and it may not obey simple scaling.
Precise statement
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.
What would settle it
An exact computation of avalanche statistics using the Abelian structure, or a proof that simple finite-size scaling fails, consistent with simulations.