Study on stochastic and quantum models on complex networks

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There are many results on two-state space homogeneous quantum walks in one dimension. Therefore we investigated localization and limit theorems for some extended models. In fact we considered the one-dimensional quantum walk with one defect. In this model, we obtained necessary and sufficient conditions of localization and the weak limit theorems. Furthermore we got the eigenvalue and eigenvector of the threshold model. By using them, we analyzed limit measures of the continuous-time quantum walk on the model and clarified difference between the quantum walk and the corresponding classical random walk.