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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.

我们使用调和分析方法研究了概率模型的渐近行为。主要结果包括：1.随机游走中的截止现象和2.代数概率中的中心极限定理。1．截止现象是马尔可夫链收敛到均衡过程中广泛观察到的一种临界现象。众所周知，由系统的对称性引起的转移矩阵特征值的重数起着重要的作用。在这个项目中，我们基于第二特征值的退化，寻求一个严格且实用的标准来解释截止现象，而这些标准无法在个体模型中进行验证，也无法基于第二特征值的退化进行直观理解。着眼于距离正则图，我们获得了用图的光谱数据描述的标准。这使我们能够系统地寻找截止现象的模型。 2．量子中心极限定理形成了代数概率的主流。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，重点关注它们的代数和组合方面。作为具体结果，我们提到了低温和无限体积极限下拉普拉斯算子在约翰逊图上相对于吉布斯态的渐近谱分布。这个结果引导我们在非平凡的相互作用的福克空间上考虑创造者和毁灭者，因此在这个方向上给出了一个很好的工作例子。