Size distribution of different types of sites in Abelian sandpile avalanches

The avalanches in the Abelian sandpile model have a compact structure and consist of shells. The different shells consist solely of sites that have toppled a fixed number of times. For the outermost shell the number is one and increases successively by one from an outer shell to the next inner shell. In this paper, we determine the size-distribution exponents of various categories of sites: sites that have toppled at least once, at least twice, at least thrice, only once and only twice. A particular subclass of avalanches is identified which have no shell structure and consist solely of sites which have toppled only once. The size distribution exponent of this type of clusters is determined.