Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model

By an inversion symmetry, we show that in the Abelian sandpile model the pr obability distribution of dissipating waves of topplings that touch the bou ndary of the system shows a power-law relationship with critical exponent 5 /8 and the probability distribution of those dissipating waves that are als o last in an avalanche has an exponent of 1. Our extensive numerical simula tions not only support these predictions, but also show that inversion symm etry is useful for the analysis of the two-wave probability distributions.