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

Abstract

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