Elephant random walk is a kind of one-dimensional discretetime random walk with infinite memory: For each step, with probability α the walker adopts one of his/her previous steps uniformly chosen at random, and otherwise he/she performs like a simple random walk (possibly with bias). It admits phase transition from diffusive to superdiffusive behavior at the critical value αc = 1/2. For α ∈ (αc, 1), there is a scaling factor an of order n α such that the position Sn of the walker at time n scaled by an converges to a nondegenerate random variable W , whose distribution is not Gaussian. Our main result shows that the fluctuation of Sn around W · an is still Gaussian. We also give a description of phase transition induced by bias decaying polynomially in time.

We attempt to analyze a one-dimensional space-inhomogeneous quantum walk (QW) with one defect at the origin, which has two different quantum coins in positive and negative parts. We call the QW ''the two-phase QW with one defect'', which we treated concerning localization theorems [7]. The two-phase QW with one defect has been expected to be a mathematical model of topological insulator [15] which is an intense issue both theoretically and experimentally [3,5,11]. In this paper, we derive the weak limit theorem describing the ballistic spreading, and as a result, we obtain the mathematical expression of the whole picture of the asymptotic behavior. Our approach is based mainly on the generating function of the weight of the passages. We emphasize that the timeaveraged limit measure is symmetric for the origin [7], however, the weak limit measure is asymmetric, which implies that the weak limit theorem represents the asymmetry of the probability distribution.

We consider stationary measures of the one-dimensional discrete-time quantum walks (QWs) with two chiralities, which is defined by a 2 $\times$ 2 unitary matrix $U$. In our previous paper \cite{Konno2014}, we proved that any uniform measure becomes the stationary measure of the QW by solving the corresponding eigenvalue problem. This paper reports that non-uniform measures are also stationary measures of the QW except when $U$ is diagonal. For diagonal matrices, we show that any stationary measure is uniform. Moreover, we prove that any uniform measure becomes a stationary measure for more general QWs not by solving the eigenvalue problem but by a simple argument.

The present paper treats the period T N of the Hadamard walk on a cycle C N with N vertices. Dukes (2014) considered the periodicity of more general quantum walks on C N and showed T 2 ¼ 2, T 4 ¼ 8, T 8 ¼ 24 for the Hadamard walk case. We prove that the Hadamard walk does not have any period except for his case, i.e., N ¼ 2; 4; 8. Our method is based on a path counting and cyclotomic polynomials which is different from his approach based on the property of eigenvalues for unitary matrix that determines the evolution of the walk.

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