Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models

本研究的主要目的是研究对称群和其他类似离散群的表示的各种特征量随着群的大小的增长的渐近行为，并从概率论和统计力学中的标度极限的观点研究极限图像。本研究的特点是利用量子概率极限定理的方法，重视与自由概率和随机矩阵的联系。以下是几项具体成果。我们研究了零温度和无限体积极限下，Laplacian函数相对于Gibbs态的谱分布，当图随着它们的度和温度保持一定的标度平衡而增长时。在量子中心极限定理的框架下，利用相互作用的Fock空间中的产生算符和湮灭算符，详细计算了系统的渐近行为.通过对Jucys-Murphy元素矩的组合硬分析，我们研究了由Young图组成的各种统计集合中浓度现象的普遍理解，包括那些来自对称群表示的不可约分解的现象，如Littlewood-Richardson系数。其中许多问题与一类修正的Young图上的随机游动的性质密切相关。在这里，我们也有效地应用了量子概率的方法.在与T. Hirai和E. Hirai，我们构造了一个很好的因子表示，它表示了以无限对称群为矩阵元的紧群的圈积的任何特征标。该表示直接反映了超出Gelfand-Raikov的一般表示的字符的特征参数。

The main purpose of the present research is to study asymptotic behavior of various characteristic quantities of representations of symmetric groups and other similar discrete groups as the sizes of the groups grow, and to investigate the limiting pictures from the viewpoint of scaling limits in probability theory and statistical mechanics. Features to be noted in this research include using methods of limit theorems in quantum probability and making much of relations to free probability and random matrices. The following are several concrete results.1. We studied the spectral distributions of Laplacians with respect to the Gibbs states in zero temperature and infinite volume limit as graphs grow with their degrees and temperatures keeping certain scaling balances. We computed the asymptotic behavior in details under the formulation of quantum central limit theorem by using creation and annihilation operators on interacting Fock spaces.2. Through combinatorial hard analysis of moments of the Jucys-Murphy element, we studied universal understanding of concentration phenomena in various statistical ensembles consisting of Young diagrams, including those which come from irreducible decomposition of a representation of the symmetric group such as the Littlewood-Richardson coefficients. Many of them are closely related to some properties of random walks on a certain modified Young graph. Here also we applied methods of quantum probability effectively.3. Under cooperation with T. Hirai and E. Hirai, we constructed a nice factor representation which expresses any character of a wreath product of a compact group with the infinite symmetric group as its matrix element. This representation reflects directly the characterizing parameters for the character beyond a general representation of Gelfand-Raikov.