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.