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

利用调和分析的方法研究了概率模型的渐近行为。主要结果包括：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在收敛到平衡点过程中广泛观察到的一种临界现象。众所周知，由系统对称性引起的转移矩阵本征值的重数起着重要的作用。在这个项目中，我们寻求一个严格而实用的判据来判断单个模型中无法验证的截断现象，并基于第二特征值的退化来直观地理解。以距离正则图为研究对象，给出了用图的谱数据来刻画距离正则图的判据。这使得我们能够系统地找到截断现象的模型。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与中心极限定理之间的重要关系，重点研究了它们的代数和组合方面。作为具体结果，我们给出了低温无限体积极限下Johnson图上拉普拉斯算符相对于Gibbs态的渐近谱分布。这一结果使我们考虑了非平凡相互作用Fock空间上的创建者和零化子，从而在这个方向上给出了一个很好的工作例子。

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.