Natural Measures on Polyominoes Induced by the Abelian Sandpile Model
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914846952390656 |
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| author | Sportiello, Andrea |
| author_facet | Sportiello, Andrea |
| contents | 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.
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| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_16418 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Natural Measures on Polyominoes Induced by the Abelian Sandpile Model Sportiello, Andrea Discrete Mathematics G.2.1; G.3 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. |
| title | Natural Measures on Polyominoes Induced by the Abelian Sandpile Model |
| topic | Discrete Mathematics G.2.1; G.3 |
| url | https://arxiv.org/abs/2406.16418 |