随机杨图极限形状的神经变分框架
Neural variational framework for random Young-diagram limit shapes
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中文总结 AI 辅助
该研究提出适配随机杨图系综结构与尺度的保持结构神经变分框架，在多类系综上验证后，用于研究无解析鞍形的四次变形钩长系综，为变形依赖的宏观鞍形族提供数值证据。
中文摘要 AI 辅助
我们针对随机杨图系综开发了一种保持结构的神经变分框架，其表示形式适配每个测度的结构与尺度。该方法在Plancherel系综、均匀系综、最小差系综和固定\boldsymbol{q}的q-Plancherel系综上得到验证，仅使用已知渐近分布用于训练后比较。随后，我们研究了无解析鞍形假设的四次变形钩长系综，将大\boldsymbol{n}神经分布与通过精确作用搜索得到的有限大小MAP分布、以及通过角转移Metropolis-Hastings采样得到的平均分布进行对比。变形增大时会抑制前导行并扩宽支撑域，且神经分布、离散分布与采样平均分布的吻合度达百分位级别。这些结果为变形依赖的宏观鞍形族提供了数值证据。
英文摘要
We develop a structure-preserving neural variational framework for random Young-diagram ensembles, with representations adapted to the structure and scaling of each measure. The method is validated on the Plancherel, uniform, minimal-difference, and fixed-\(q\) \(q\)-Plancherel ensembles, using known asymptotic profiles only for post-training comparison. We then study a quartically deformed hook-length ensemble without assuming an analytical saddle shape. Large-\(n\) neural profiles are compared with finite-size MAP profiles obtained from exact-action searches and with mean profiles obtained from corner-transfer Metropolis--Hastings sampling. Increasing the deformation suppresses the leading rows and broadens the support, while the neural, discrete, and sampled mean profiles agree at the percent level. These results provide numerical evidence for a deformation-dependent macroscopic saddle family.
