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

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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.