WAVES OF TOPPLINGS IN AN ABELIAN SANDPILE

We study the structure of the avalanche process in the 2D Abelian sandpile model. An avalanche may be represented as a sequence of waves, each consisting of sites that toppled only once in that wave. It is shown that the waves are in one-to-one correspondence with the set of two-rooted spanning trees and can be described in terms of Green functions. We argue that the probability distribution of waves of size s varies as 1/s for large s. We prove also that the avalanches started at the open boundary consist of only one wave and find the asymptotic distribution of their sizes, which varies as s(-3/2). We establish the equivalence between waves and inverse avalanches introduced by Dhar and Manna and reproduce their result 11/8 for the critical exponent of the size distribution of the first inverse avalanche or, in our terms, the last wave.