Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908471183540224 |
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| author | Knizel, Alisa Petrov, Leonid |
| author_facet | Knizel, Alisa Petrov, Leonid |
| contents | We investigate the asymptotic behavior of the q-Racah probability measure on lozenge tilings of a hexagon whose side lengths scale linearly with a large parameter $L$, while the parameters $q\in(0,1)$ and $κ\in \mathbf{i}\mathbb{R}$ remain fixed. This regime differs fundamentally from the traditional case $q\sim e^{-c/L}\to1$, in which random tilings are locally governed by two-dimensional translation-invariant ergodic Gibbs measures. In the fixed-q regime we uncover a new macroscopic phase, the waterfall (previously only observed experimentally), where the two-dimensional Gibbs structure collapses into a one-dimensional random stepped interface that we call a barcode. We prove a law of large numbers and exponential concentration, showing that the random tilings converge to a deterministic waterfall profile. We further conjecture an explicit correlation kernel of the one-dimensional barcode process arising in the limit. Remarkably, the limit is invariant under shifts by $2\mathbb{Z}$ but not by $\mathbb{Z}$, exhibiting an emergent period-two structure absent from the original weights. Our conjectures are supported by extensive numerical evidence and perfect sampling simulations. The kernel is built from a family of functions orthogonal in both spaces $\ell^{2}(\mathbb{Z})$ and $\ell^{2}(\mathbb{Z}+\frac12)$, that may be of independent interest. Our proofs adapt the spectral projection method of Borodin-Gorin-Rains (arXiv:0905.0679) to the regime with fixed~q. The resulting asymptotic analysis is substantially more involved, and leads to non-self-adjoint operators. We overcome these challenges in the exponential concentration result by a separate argument based on sharp bounds for the ratios of probabilities under the q-Racah orthogonal polynomial ensemble. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_22011 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures Knizel, Alisa Petrov, Leonid Probability Statistical Mechanics Mathematical Physics 60K35, 60G55, 33D45, 82B41, 82B23 We investigate the asymptotic behavior of the q-Racah probability measure on lozenge tilings of a hexagon whose side lengths scale linearly with a large parameter $L$, while the parameters $q\in(0,1)$ and $κ\in \mathbf{i}\mathbb{R}$ remain fixed. This regime differs fundamentally from the traditional case $q\sim e^{-c/L}\to1$, in which random tilings are locally governed by two-dimensional translation-invariant ergodic Gibbs measures. In the fixed-q regime we uncover a new macroscopic phase, the waterfall (previously only observed experimentally), where the two-dimensional Gibbs structure collapses into a one-dimensional random stepped interface that we call a barcode. We prove a law of large numbers and exponential concentration, showing that the random tilings converge to a deterministic waterfall profile. We further conjecture an explicit correlation kernel of the one-dimensional barcode process arising in the limit. Remarkably, the limit is invariant under shifts by $2\mathbb{Z}$ but not by $\mathbb{Z}$, exhibiting an emergent period-two structure absent from the original weights. Our conjectures are supported by extensive numerical evidence and perfect sampling simulations. The kernel is built from a family of functions orthogonal in both spaces $\ell^{2}(\mathbb{Z})$ and $\ell^{2}(\mathbb{Z}+\frac12)$, that may be of independent interest. Our proofs adapt the spectral projection method of Borodin-Gorin-Rains (arXiv:0905.0679) to the regime with fixed~q. The resulting asymptotic analysis is substantially more involved, and leads to non-self-adjoint operators. We overcome these challenges in the exponential concentration result by a separate argument based on sharp bounds for the ratios of probabilities under the q-Racah orthogonal polynomial ensemble. |
| title | Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures |
| topic | Probability Statistical Mechanics Mathematical Physics 60K35, 60G55, 33D45, 82B41, 82B23 |
| url | https://arxiv.org/abs/2507.22011 |