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

图和离散组的调和分析以及概率模型的缩放限制

基本信息

批准号：
13640175
负责人：
HORA Akihito
金额：
$ 2.24万
依托单位：
Okayama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640175/
关键词：
spectrum of graph scaling limit central limit theorem distance-regular graph representation of symmetric group quantum probability harmonic analysis method of quantum decomposition ヤング図形代数的確率論対称群相互作用フォック空間確率モデル

项目摘要

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.

本课题的目的是利用调和分析和表示理论的方法，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大系统的统计性质。我们取得了如下的具体成果。在量子中心极限定理的框架下，计算了图上邻接算子谱分布的尺度极限。我们在距离正则图上引入了Gibbs态和真空态，研究了低温和高温下Johnson图的极限图。结果用与梅克纳多项式相关的相互作用的Fock空间来描述。利用创造者和湮灭者的组合结构，在极限下得到了有趣的分布。用量子分解的方法建立了图谱分析的一般理论。我们揭示了正则图的渐近特征值与相互作用的Fock空间参数之间的联系。在个别谱限计算之外，利用正交多项式和格林函数的方法系统地描述了谱限分布。这里的项目与东北大学与Obata Nobuaki的合作密切相关。我们得到了不可约特征的Kerov中心极限定理的一个推广（量子化）和作为对称群表示的渐近方面的Plancherel测度。由于结果超出了相互作用的Fock空间的框架，我们引入了对通常的Young图的修改以及其上的创造者和湮灭者。