Natural Measures on Polyominoes Induced by the Abelian Sandpile Model
<|reference_start|>Natural Measures on Polyominoes Induced by the Abelian Sandpile Model: 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.<|reference_end|>
@article{sportiello2024natural,
    title={Natural Measures on Polyominoes Induced by the Abelian Sandpile Model},
    author={Andrea Sportiello (LIPN, and CNRS, Universit'e Sorbonne Paris Nord)},
    journal={EPTCS 403, 2024, pp. 177-190},
    year={2024},
    doi={10.4204/EPTCS.403.35},
    archivePrefix={arXiv},
    eprint={2406.16418},
    primaryClass={cs.DM}
}