Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908446574510080 |
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| author | Bufetov, Alexey Petrov, Leonid Zografos, Panagiotis |
| author_facet | Bufetov, Alexey Petrov, Leonid Zografos, Panagiotis |
| contents | We study the asymptotic behavior of random domino tilings of the Aztec diamond of size $M$ in a random environment, where the environment is a one-periodic sequence of i.i.d. random weights attached to domino positions (i.e., to the edges of the underlying portion of the square grid). We consider two cases: either the variance of the weights decreases at a critical scale $1/M$, or the distribution of the weights is fixed. In the former case, the unrescaled fluctuations of the domino height function are governed by the sum of a Gaussian Free Field and an independent Brownian motion. In the latter case, we establish fluctuations on the much larger scale $\sqrt M$, given by the Brownian motion alone.
To access asymptotic fluctuations in random environment, we employ the method of Schur generating functions. Moreover, we substantially extend the known Law of Large Numbers and Central Limit Theorems for particle systems via Schur generating functions in order to apply them to our setting. These results might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_08560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions Bufetov, Alexey Petrov, Leonid Zografos, Panagiotis Probability Statistical Mechanics Mathematical Physics Combinatorics 60K35, 82B20, 05E05, 60C05, 60F05, 82B44 We study the asymptotic behavior of random domino tilings of the Aztec diamond of size $M$ in a random environment, where the environment is a one-periodic sequence of i.i.d. random weights attached to domino positions (i.e., to the edges of the underlying portion of the square grid). We consider two cases: either the variance of the weights decreases at a critical scale $1/M$, or the distribution of the weights is fixed. In the former case, the unrescaled fluctuations of the domino height function are governed by the sum of a Gaussian Free Field and an independent Brownian motion. In the latter case, we establish fluctuations on the much larger scale $\sqrt M$, given by the Brownian motion alone. To access asymptotic fluctuations in random environment, we employ the method of Schur generating functions. Moreover, we substantially extend the known Law of Large Numbers and Central Limit Theorems for particle systems via Schur generating functions in order to apply them to our setting. These results might be of independent interest. |
| title | Domino Tilings of the Aztec Diamond in Random Environment and Schur Generating Functions |
| topic | Probability Statistical Mechanics Mathematical Physics Combinatorics 60K35, 82B20, 05E05, 60C05, 60F05, 82B44 |
| url | https://arxiv.org/abs/2507.08560 |