Boundary dents, the arctic circle and the arctic ellipse ...
The original motivation for this paper goes back to the mid-1990's, when James Propp was interested in natural situations when the number of domino tilings of a region increases if some of its unit squares are deleted. Guided in part by the intuition one gets from earlier work on parallels betw...
| Main Authors: | , |
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| Format: | Article in Journal/Newspaper |
| Language: | unknown |
| Published: |
arXiv
2023
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| Subjects: | |
| Online Access: | https://dx.doi.org/10.48550/arxiv.2308.06863 https://arxiv.org/abs/2308.06863 |
| Summary: | The original motivation for this paper goes back to the mid-1990's, when James Propp was interested in natural situations when the number of domino tilings of a region increases if some of its unit squares are deleted. Guided in part by the intuition one gets from earlier work on parallels between the number of tilings of a region with holes and the 2D Coulomb energy of the corresponding system of electric charges, we consider Aztec diamond regions with unit square defects along two adjacent sides. We show that for large regions, if these defects are at fixed distances from a corner, the ratio between the number of domino tilings of the Aztec diamond with defects and the number of tilings of the entire Aztec diamond approaches a Delannoy number. When the locations of the defects are not fixed but instead approach given points on the boundary of the scaling limit $S$ (a square) of the Aztec diamonds, we prove that, provided the line segment connecting these points is outside the circle inscribed in $S$, this ... : 38 pages ... |
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