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

本项目的目的是从渐近谱分析和标度极限的观点出发，利用调和分析和表示论的方法，读出具有一定对称性的巨系统的统计性质。我们得到了以下具体结果。1.在量子中心极限定理的框架下，我们计算了图上邻接算子谱分布的标度极限。在距离正则图上引入了吉布斯态和真空态，并研究了低温和高温下的极限图，特别是约翰逊图的极限图.结果描述的相互作用Fock空间与Meixner多项式。利用创造子和零化子的组合结构，在极限下得到了有趣的分布.本文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Fock空间参数之间的关系。利用正交多项式和绿色函数的方法，系统地描述了极限分布。这里的项目与东北大学Nobuaki Obata的合作密切相关。我们得到了一个扩展（量化）Kerov的中心极限定理的不可约字符和Plancherel措施作为一个渐近方面的陈述的对称群。由于这个结果超出了相互作用Fock空间的框架，我们引入了通常的Young图的一个修改以及它的生成子和零化子。

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.