Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt3$)
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911616749010944 |
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| author | Gao, Haitao Bharadwaj, Aaryash |
| author_facet | Gao, Haitao Bharadwaj, Aaryash |
| contents | The Smith Hat tile is the first known aperiodic monotile, having been discovered in 2023. The simple structure, constructed using only 8 kites, is unique and well motivated for analysis within percolation theory. The primary goal of this paper is to discover the critical threshold $p_c$ in both site and bond Bernoulli structures using Monte Carlo simulation for the Smith hat tile(1,$\sqrt3$). Our findings are site and bond values of $p_c^s = 0.822725 \pm 0.000044$ and $p_c^b = 0.798161 \pm 0.000044$ for edge percolation and $0.544247 \pm 0.000101$ for site percolation on the dual graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt3$) Gao, Haitao Bharadwaj, Aaryash Statistical Mechanics Data Analysis, Statistics and Probability The Smith Hat tile is the first known aperiodic monotile, having been discovered in 2023. The simple structure, constructed using only 8 kites, is unique and well motivated for analysis within percolation theory. The primary goal of this paper is to discover the critical threshold $p_c$ in both site and bond Bernoulli structures using Monte Carlo simulation for the Smith hat tile(1,$\sqrt3$). Our findings are site and bond values of $p_c^s = 0.822725 \pm 0.000044$ and $p_c^b = 0.798161 \pm 0.000044$ for edge percolation and $0.544247 \pm 0.000101$ for site percolation on the dual graph. |
| title | Percolation Critical Probability of Aperiodic Smith Hat tile(1, $\sqrt3$) |
| topic | Statistical Mechanics Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2604.21165 |