Formation of Avalanches and Critical Exponents in an Abelian Sandpile Model

Abstract

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 $\ensuremath{\alpha}$ simply related to the basic exponent $\ensuremath{\tau}$ of the sandpile model. Using known exponents for the spanning trees, we derive from scaling arguments $\ensuremath{\alpha}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3/4$ and $\ensuremath{\tau}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}5/4$.