On loops in the complement to dimers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909430548791296 |
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| author | Glazman, Alexander Rey, Lucas |
| author_facet | Glazman, Alexander Rey, Lucas |
| contents | We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop $O(1)$ model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path.
Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_11708 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On loops in the complement to dimers Glazman, Alexander Rey, Lucas Probability 82B20 (Primary) 82b43, 60K35 (Secondary) We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop $O(1)$ model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path. Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path. |
| title | On loops in the complement to dimers |
| topic | Probability 82B20 (Primary) 82b43, 60K35 (Secondary) |
| url | https://arxiv.org/abs/2412.11708 |