Harmonic analysis on graphs and discrete groups, and scaling limit for probability models

本项目的目的是利用调和分析和表象理论的方法，从渐近谱分析和标度极限的角度读出具有一定对称性的巨系统的统计性质。取得的具体成果如下：1.在量子中心极限定理的框架下，计算了图上邻接算子的谱分布的标度极限。我们在距离正则图上引入了Gibbs态和真空态，并研究了低温和高度下的极限图，特别是Johnson图。结果用与Meixner多项式相关的相互作用Fock空间来描述。利用创建者和零化者的组合结构，在极限内得到了有趣的分布。我们用量子分解的方法建立了图的谱分析的一般理论。揭示了正则图的渐近特征值与相互作用的Fock空间的参数之间的关系。利用正交多项式和格林函数的方法，系统地描述了超越单个谱极限计算的极限分布。这里的项目与与东北大学小田信明的一项合作密切相关。我们得到了关于不可约特征标的Kerov中心极限定理的一个推广(量子化)，以及作为对称群表示的渐近方面的Plancerel测度。由于结果脱离了交互Fock空间的框架，我们引入了对通常的Young图的一个修改，以及它上的创建子和零化子。

The aim of this project is to read out statistical properties of huge systems characterized by a certain symmetry from the viewpoint of asymptotic spectral analysis and scaling limits by using the methods of harmonic analysis and representation theory. We obtained concrete results as follows.1. We computed scaling limits for the spectral distributions of adjacency operators on graphs in the framework of quantum central limit theorem. We introduced Gibbs states as well as vacuum states on distance-regular graphs and investigated the limit picture in low temperature and high degrees especially for Johnson graphs. The result is described in terms of the interacting Fock space associated with Meixner polynomials. Interesting distributions are derived in the limit by using combinatorial structure of creators and annihilators.2. We established a general theory for spectral analysis of graphs by the method of quantum decomposition. We revealed a connection of asymptotic characteristic values of regular graphs with the parameters of interacting Fock spaces. The limit distributions are systematically described by using methods of orthogonal polynomials and Green functions beyond computation of individual spectral limits. The item here is closely related to a joint work with Nobuaki Obata at Tohoku University.3. We obtained an extension (a quantization) of Kerov's central limit theorem for irreducible characters and the Plancherel measure as an asymptotic aspect of representations of the symmetric groups. Since the result goes out of the framework of interacting Fock spaces, we introduced a modification of the usual Young graph as well as creators and annihilators on it.