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:
  • 金额:
    $ 1.86万
  • 依托单位:
  • 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

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。

项目成果

期刊论文数量(7)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
A. Hora: "Gibbs state on a distance-regular graph and its application to a scaling limit of the spectral distributions of discrete Laplacians"Probability Theory and Related Fields. (To appear).
A. Hora:“距离正则图上的吉布斯状态及其在离散拉普拉斯谱分布的缩放极限中的应用”概率论和相关领域。
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A. Hora: "An axiomatic approach to the cut-off phenomenon for random walks on large distance-regular graphs"Hiroshima Mathematical Journal. (To appear).
A. Hora:“大距离正则图上随机游走截止现象的公理化方法”广岛数学杂志。
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A. Hora: "Scaling limit of the spectral distributions of the Laplacions on large graphs"Transactions of German-Japanese Symposium 1999 in Kyoto. (To appear).
A. Hora:“大图上拉普拉西翁光谱分布的标度限制”1999 年京都德日研讨会论文集。
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Hora,A.: "An axiomatic approach to the cut-off phenomenon for random walks on large distance-regular graphs"Hiroshima Math.J.. 30. 271-299 (2000)
Hora,A.:“大距离正则图上随机游走截止现象的公理化方法”Hiroshima Math.J.. 30. 271-299 (2000)
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Hora,A.: "Gibbs state, quadratic embedding, and central limit theorem on large graphs"Quantum Information III. (in press).
Hora,A.:“大图上的吉布斯态、二次嵌入和中心极限定理”量子信息 III。
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HORA Akihito其他文献

HORA Akihito的其他文献

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{{ truncateString('HORA Akihito', 18)}}的其他基金

Study on decomposition of unitary representations of groups and harmonic functions on branching graphs
分支图上群和调和函数的酉表示分解研究
  • 批准号:
    23540197
  • 财政年份:
    2011
  • 资助金额:
    $ 1.86万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
Integrated study of probability and representation theory towards harmonic analysis on huge groups
大群调和分析的概率与表示论综合研究
  • 批准号:
    19340032
  • 财政年份:
    2007
  • 资助金额:
    $ 1.86万
  • 项目类别:
    Grant-in-Aid for Scientific Research (B)
Study of asymptotic theory for representations of symmetric groups from the viewpoint of scaling limits for probability models
从概率模型标度极限的角度研究对称群表示的渐近理论
  • 批准号:
    16540154
  • 财政年份:
    2004
  • 资助金额:
    $ 1.86万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)
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
  • 批准号:
    13640175
  • 财政年份:
    2001
  • 资助金额:
    $ 1.86万
  • 项目类别:
    Grant-in-Aid for Scientific Research (C)

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