Time-inhomogeneous Fokker-Planck equation for wave distributions in the Abelian sandpile model

The time and size distributions of the waves of topplings in the Abelian sa ndpile model are expressed as the first arrival at the origin distribution for a scale invariant, time-inhomogeneous Fokker-Plank equation. Assuming a linear conjecture for the time inhomogeneity exponent as a function of a l oop-erased random walk (LERW) critical exponent, suggested by numerical res ults, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three di mensions. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.