Formation of Avalanches and Critical Exponents in an Abelian Sandpile Model

The structure of avalanches in the Abelian sandpile model on a square lattice is analyzed. It is shown that an avalanche can be considered as a sequence of waves of decreasing sizes. Being more simple objects, waves admit a representation in terms of spanning trees covering the lattice sites. The difference in sizes of subsequent waves follows a power law with the exponent $\alpha$ simply related to the basic exponent $\tau$ of the sandpile model. Using known exponents for the spanning trees, we derive from scaling arguments $\alpha =3/4\mathrm{}$ and $\tau =5/4\mathrm{}$.