Critical exponents for boundary avalanches in two-dimensional Abelian sandpile

Abstract

We investigate the properties of boundary avalanches in 2D Abelian sandpile model (ASM). We construct the one-to-one correspondence of boundary avalanches and two-rooted spanning trees. Using the connection between the obtained graph representation and lattice Green functions we calculate the exact values of critical exponents for size and lifetime distributions of avalanches starting at the open boundary that forms an angle alpha . We find that the probability of a boundary avalanche of the size s varies as s-1- pi 2 alpha / for large s and the probability of an avalanche of lifetime t varies as t-1-4 pi 5 alpha / for large t. The obtained values are verified by numerical simulations.