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Harmonic analysis on discrete structures and its applications to classical and quantum probability models

离散结构的调和分析及其在经典和量子概率模型中的应用

基本信息

批准号：
11640168
负责人：
HORA Akihito
金额：
$ 1.86万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640168/
关键词：
harmonic analysis probability model random walk quantum probability central limit theorem spectrum distance-regular graph the cut-off phenomenon 対称群スケーリング極限

项目摘要

We studied asymptotic behavior of probability models by using the methods of harmonic analysis. Main results are included in 1. the cut-off phenomenon in random walks and 2. central limit theorems in algebraic probability.1. The cut-off phenomenon is a sort of critical phenomenon widely observed in the process of convergence to the equilibrium for Markov chains. It is known that the multiplicities of eigenvalues of a transition matrix, which are caused by symmetry of the system, play an important role. In this project, we seeked a rigorous and practical criterion for the cut-off phenomenon beyond verification in individual models and intuitive understanding based on the degeneration of the second eigenvalue. Focusing on distance-regular graphs, we obtained a criterion described in terms of spectral data of the graph. This enables us to find models of the cut-off phenomenon systematically.2. Quantum central limit theorems form a main stream in algebraic probability. In this project, we studied important relations between independence of noncommutative random variables and central limit theorems, making much of their algebraic and combinatorial aspects. As a concrete result, we mention the asymptotic spectral distribution of the Laplacian operator on a Johnson graph with respect to the Gibbs state under a low temperature and infinite volume limit. This result leads us to the consideration of creators and annihilators on a nontrivial interacting Fock space and hence gives a good working example in this direction.

利用调和分析的方法研究了概率模型的渐近性态。主要结果包括在1。随机游动中的截断现象; 2.代数概率中的中心极限定理。截断现象是马尔可夫链收敛到平衡点过程中普遍存在的一种临界现象。由系统对称性引起的转移矩阵特征值的重数对系统的稳定性起着重要的作用。在这个项目中，我们寻求一个严格的和实用的标准，在个别模型的验证和直观的理解的基础上的第二个特征值的退化的截止现象。针对距离正则图，我们得到了一个用图的谱数据描述的判别准则。这使我们能够系统地找到截止现象的模型。量子中心极限定理是代数概率论的主流。在这个项目中，我们研究了非交换随机变量的独立性和中心极限定理之间的重要关系，使它们的代数和组合方面。作为一个具体的结果，我们提到了在低温和无限体积极限下，拉普拉斯算子在约翰逊图上关于吉布斯态的渐近谱分布。这个结果使我们考虑的创造者和零化子的非平凡的相互作用的Fock空间，因此在这个方向上提供了一个很好的工作例子。