Perfect t-embeddings and Lozenge Tilings
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866913464981651456 |
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| author | Berggren, Tomas Nicoletti, Matthew Russkikh, Marianna |
| author_facet | Berggren, Tomas Nicoletti, Matthew Russkikh, Marianna |
| contents | We construct perfect t-embeddings for regular hexagons of the hexagonal lattice, providing the first example, and hence proving existence, for graphs with an outer face of degree greater than four. The construction is in terms of the inverse Kasteleyn matrix and relies only on symmetries of the graph. Using known formulas for the inverse Kasteleyn matrix, we derive exact contour integral formulas for these embeddings and their origami maps. Through steepest descent analysis, we establish scaling limits, proving convergence of origami maps to a maximal surface in the Minkowski space $\mathbb{R}^{2,1}$, and we verify structural rigidity conditions, leading to a new proof of convergence of height fluctuations to the Gaussian free field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_05441 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perfect t-embeddings and Lozenge Tilings Berggren, Tomas Nicoletti, Matthew Russkikh, Marianna Probability Mathematical Physics Combinatorics 82B20 We construct perfect t-embeddings for regular hexagons of the hexagonal lattice, providing the first example, and hence proving existence, for graphs with an outer face of degree greater than four. The construction is in terms of the inverse Kasteleyn matrix and relies only on symmetries of the graph. Using known formulas for the inverse Kasteleyn matrix, we derive exact contour integral formulas for these embeddings and their origami maps. Through steepest descent analysis, we establish scaling limits, proving convergence of origami maps to a maximal surface in the Minkowski space $\mathbb{R}^{2,1}$, and we verify structural rigidity conditions, leading to a new proof of convergence of height fluctuations to the Gaussian free field. |
| title | Perfect t-embeddings and Lozenge Tilings |
| topic | Probability Mathematical Physics Combinatorics 82B20 |
| url | https://arxiv.org/abs/2408.05441 |