Counting domino and lozenge tilings of reduced domains with Padé-type approximants
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| Format: | Preprint |
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2025
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| _version_ | 1866908796990783488 |
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| author | Charlier, Christophe Claeys, Tom |
| author_facet | Charlier, Christophe Claeys, Tom |
| contents | We introduce a new method for studying gap probabilities in a class of discrete determinantal point processes with double contour integral kernels. This class of point processes includes uniform measures of domino and lozenge tilings as well as their doubly periodic generalizations. We use a Fourier series approach to simplify the form of the kernels and to characterize gap probabilities in terms of Riemann-Hilbert problems.
As a first illustration of our approach, we obtain an explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Padé approximants, by solving the associated Riemann-Hilbert problem explicitly. As a second application, we obtain an explicit expression for the number of lozenge tilings of (simply connected) reduced hexagons in terms of Hermite-Padé approximants. For more complicated domains, such as hexagons with holes, the number of tilings involves a generalization of Hermite-Padé approximants. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19096 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counting domino and lozenge tilings of reduced domains with Padé-type approximants Charlier, Christophe Claeys, Tom Probability Classical Analysis and ODEs We introduce a new method for studying gap probabilities in a class of discrete determinantal point processes with double contour integral kernels. This class of point processes includes uniform measures of domino and lozenge tilings as well as their doubly periodic generalizations. We use a Fourier series approach to simplify the form of the kernels and to characterize gap probabilities in terms of Riemann-Hilbert problems. As a first illustration of our approach, we obtain an explicit expression for the number of domino tilings of reduced Aztec diamonds in terms of Padé approximants, by solving the associated Riemann-Hilbert problem explicitly. As a second application, we obtain an explicit expression for the number of lozenge tilings of (simply connected) reduced hexagons in terms of Hermite-Padé approximants. For more complicated domains, such as hexagons with holes, the number of tilings involves a generalization of Hermite-Padé approximants. |
| title | Counting domino and lozenge tilings of reduced domains with Padé-type approximants |
| topic | Probability Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2502.19096 |