Natural Measures on Polyominoes Induced by the Abelian Sandpile Model

TL;DR

This paper introduces a natural measure on polyominoes derived from boundary avalanches in the Abelian Sandpile Model, suggesting a power-law distribution for avalanche sizes with an exponent of 3/2, supported by numerical evidence.

Contribution

It proposes a new probabilistic measure on polyominoes based on boundary avalanches, providing insights into avalanche size distributions and statistical properties within the Abelian Sandpile Model.

Findings

Avalanche size distribution follows a power-law with exponent 3/2.

Numerical evidence supports the proposed distribution and measure.

Analyzed statistical observables, including the density of triple points.

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.