随机菱形瀑布：Gibbs测度的维数坍缩
Random Lozenge Waterfall: Dimensional Collapse of Gibbs Measures
AI总结：
该研究在固定q参数下分析六边形菱形铺砌的q-Racah测度渐近行为，发现瀑布相的二维Gibbs结构坍缩为一维条形码界面，证明大数定律与指数集中性，提出相关核猜想并给出数值支持。
摘要：
我们研究六边形菱形铺砌上q-Racah概率测度的渐近行为，六边形的边长随大参数$L$线性缩放，而参数$q\in(0,1)$和$\kappa\in \mathbf{i}\mathbb{R}$保持固定。该机制与传统的$q\sim e^{-c/L}\to1$情形存在本质区别，传统情形下随机铺砌的局部行为由二维平移不变遍历Gibbs测度支配。在固定q的机制下，我们发现了一种新的宏观相——瀑布相（此前仅在实验中观测到），其中二维Gibbs结构坍缩为一维随机阶梯界面，我们称之为条形码。我们证明了大数定律和指数集中性，表明随机铺砌收敛于确定性的瀑布轮廓。我们进一步推测，极限中出现的一维条形码过程存在显式相关核。值得注意的是，该极限在$2\mathbb{Z}$平移下不变，但在$\mathbb{Z}$平移下不不变，呈现出原始权重中不存在的涌现周期二结构。我们的推测得到了大量数值证据和完美采样模拟的支持。该相关核由一族在$\ell^{2}(\mathbb{Z})$和$\ell^{2}(\mathbb{Z}+\frac12)$空间中均正交的函数构造，这族函数本身可能具有独立研究价值。我们的证明将Borodin-Gorin-Rains（arXiv:0905.0679）的谱投影方法适配到固定q的机制下。由此产生的渐近分析要复杂得多，并涉及非自伴算子。我们通过基于q-Racah正交多项式系综下概率比值的精确界的独立论证，在指数集中性结果中克服了这些挑战。
英文摘要：
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.
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