Natural Measures on Polyominoes Induced by the Abelian Sandpile Model

Abstract: We introduce a natural Boltzmann measure over polyominoes induced by boundary avalanches in the Abelian Sandpile Model. Through the study of a suitable associated process, we give an argument suggesting that the probability distribution of the avalnche sizes has a power-law decay with exponent 3/2, in contrast with the present understanding of bulk avalanches in the model (which has some exponent between 1 and 5/4), and to the ordinary generating function of polyominoes (which is conjectured to have a logarithmic singularity, i.e. exponent 1). We provide some numerical evidence for our claims, and evaluate some other statistical observables on our process, most notably the density of triple points.