We establish the universal edge scaling limit of random partitions with the infinite-parameter distribution called the Schur measure. We explore the asymptotic behavior of the wave function, which is a building block of the corresponding kernel, based on the Schrödinger-type differential equation. We show that the wave function is in general asymptotic to the Airy function and its higher-order analogs in the edge scaling limit. We construct the corresponding higher-order Airy kernel and the Tracy–Widom distribution from the wave function in the scaling limit and discuss its implication to the multicritical phase transition in the large-size matrix model. We also discuss the limit shape of random partitions through the semi-classical analysis of the wave function.

我们使用称为Schur测度的无限参数分布建立随机分区的通用边缘缩放限制。我们基于Schrödinger型微分方程探索了波动函数的渐近行为，波动函数是相应内核的构建块。我们表明，在边缘缩放极限中，波动函数通常比Airy函数及其高阶模拟渐近。我们根据缩放比例范围内的波动函数构造了相应的高阶Airy核和Tracy-Widom分布，并讨论了其对大尺寸矩阵模型中多临界相变的影响。我们还通过波动函数的半经典分析来讨论随机分区的极限形状。