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Spectral and asymptotic properties of Grover walks on crystal lattice
We propose a twisted Szegedy walk for estimating the limit behavior of a discrete-time quantum walk on a crystal lattice, an infinite abelian covering graph, whose notion was introduced by [14].Expand
Combinatorial curvature for planar graphs
Regarding an infinite planar graph G as a discrete analogue of a noncompact simply connected Riemannian surface, we introduce the combinatorial curvature of G corresponding to the sectional curvatureExpand
Combinatorial curvature for planar graphs
The aim of this expository article is to exhibit several interesting interactions among geometry, graph theory and probability through a brief survey of a series of our recent work on geometry ofExpand
Spectral structure of the Laplacian on a covering graph
We study the spectral structure of the discrete Laplacian on an infinite graph and show that it has band structure and no eigenvalues. Expand
Dynamical system induced by quantum walk.
We consider the Grover walk model on a connected finite graph with two infinite length tails and we set an $\ell^\infty$-infinite external source from one of the tails as the initial state. We showExpand
Isoperimetric Constants of (d,f) Regular Planar Graphs
For a (d,f)-regular planar graph, which is an infinite planar graph embedded in the plane such that the degree of each vertex is d and the degree of each face is f, we determine two kinds ofExpand
Quantum walks induced by Dirichlet random walks on infinite trees
We consider the Grover walk on infinite trees from the view point of spectral analysis. From the previous works, infinite regular trees provide localization. In this paper, we give the completeExpand