Norio Konno

Stochastic partial differential equations for some measure-valued diffusions

SummaryWe consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R1, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.

Quantum Random Walks in One Dimension

AbstractThis letter treats the quantum random walk on the line determined by a 2 × 2 unitary matrix U. A combinatorial expression for the mth moment of the quantum random walk is presented by using 4 matrices, P, Q, R and S given by U. The dependence of the mth moment on U and initial qubit state ϕ is clarified. A new type of limit theorems for the quantum walk is given. Furthermore necessary and sufficient conditions for symmetry of distribution for the quantum walk is presented. Our results show that the behavior of quantum random walk is striking different from that of the classical ramdom walk. PACS: 03.67.Lx; 05.40.Fb; 02.50.Cw

One-dimensional three-state quantum walk

We study a generalized Hadamard walk in one dimension with three inner states. The particle governed by the three-state quantum walk moves, in superposition, both to the left and to the right according to the inner state. In addition to these two degrees of freedom, it is allowed to stay at the same position. We calculate rigorously the wave function of the particle starting from the origin for any initial qubit state and show the spatial distribution of probability of finding the particle. In contrast with the Hadamard walk with two inner states on a line, the probability of finding the particle at the origin does not converge to zero even after infinite time steps except special initial states. This implies that the particle is trapped near the origin after a long time with high probability.

Localization of two-dimensional quantum walks

The Grover walk, which is related to Grovers search algorithm on a quantum computer, is one of the typical discrete time quantum walks. However, a localization of the two-dimensional Grover walk starting from a fixed point is strikingly different from other types of quantum walks. The present paper explains the reason why the walker who moves according to the degree-four Grover operator can remain at the starting point with a high probability. It is shown that the key factor for the localization is due to the degeneration of eigenvalues of the time evolution operator. In fact, the global time evolution of the quantum walk on a large lattice is mainly determined by the degree of degeneration. The dependence of the localization on the initial state is also considered by calculating the wave function analytically.

Geographical threshold graphs with small-world and scale-free properties

Many real networks are equipped with short diameters, high clustering, and power-law degree distributions. With preferential attachment and network growth, the model by Barabási and Albert simultaneously reproduces these properties, and geographical versions of growing networks have also been analyzed. However, nongrowing networks with intrinsic vertex weights often explain these features more plausibly, since not all networks are really growing. We propose a geographical nongrowing network model with vertex weights. Edges are assumed to form when a pair of vertices are spatially close and/or have large summed weights. Our model generalizes a variety of models as well as the original nongeographical counterpart, such as the unit disk graph, the Boolean model, and the gravity model, which appear in the contexts of percolation, wire communication, mechanical and solid physics, sociology, economy, and marketing. In appropriate configurations, our model produces small-world networks with power-law degree distributions. We also discuss the relation between geography, power laws in networks, and power laws in general quantities serving as vertex weights.

Upper bounds for survival probability of the contact process

A precise description of the nontrivial upper invariant measure for λ>λc is still an open problem for the basic contact process, which is a self-dual, attractive, but nonreversible Markov process of an interacting particle system. By its self-duality, to identify the invariant measure is equivalent to determining the initial-state dependence of the survival probability of the process. A procedure to give rigorous upper bounds for the survival probability is presented based on a lemma given by Harris. Two new bounds are given, improving the simple branching-process bound. In the one-dimensional case, the present procedure can be viewed as a trial to make approximate measures by generalized Markov extensions.

Localization of an inhomogeneous discrete-time quantum walk on the line

We investigate a space-inhomogeneous discrete-time quantum walk in one dimension. We show that the walk exhibits localization by a path counting method.

One-dimensional discrete-time quantum walks on random environments

We consider discrete-time nearest-neighbor quantum walks on random environments in one dimension. Using the method based on a path counting, we present both quenched and annealed weak limit theorems for the quantum walk.

Limit Theorem for Continuous-Time Quantum Walk on the Line

Concerning a discrete-time quantum walk X(t)(d) with a symmetric distribution on the line, whose evolution is described by the Hadamard transformation, it was proved by the author that the following weak limit theorem holds: X(t)(d)/t -->dx/pi(1-x2) square root of (1-2x2) as t --> infinity. The present paper shows that a similar type of weak limit theorem is satisfied for a continuous-time quantum walk X((c) )(t ) on the line as follows: X(t)(c)/t --> dx/pi square root of (1-x2) as t --> infinity. These results for quantum walks form a striking contrast to the central limit theorem for symmetric discrete- and continuous-time classical random walks: Y(t)/square root of (t) --> e(-x2/2)dx/square root of (2pi) as t --> infinity. The work deals also with the issue of the relationship between discrete and continuous-time quantum walks. This topic, subject of a long debate in the previous literature, is treated within the formalism of matrix representation and the limit distributions are exhaustively compared in the two cases.

Transmission of severe acute respiratory syndrome in dynamical small-world networks

The outbreak of severe acute respiratory syndrome (SARS) is still threatening the world because of a possible resurgence. In the current situation that effective medical treatments such as antiviral drugs are not discovered yet, dynamical features of the epidemics should be clarified for establishing strategies for tracing, quarantine, isolation, and regulating social behavior of the public at appropriate costs. Here we propose a network model for SARS epidemics and discuss why superspreaders emerged and why SARS spread especially in hospitals, which were key factors of the recent outbreak. We suggest that superspreaders are biologically contagious patients, and they may amplify the spreads by going to potentially contagious places such as hospitals. To avoid mass transmission in hospitals, it may be a good measure to treat suspected cases without hospitalizing them. Finally, we indicate that SARS probably propagates in small-world networks associated with human contacts and that the biological nature of individuals and social group properties are factors more important than the heterogeneous rates of social contacts among individuals. This is in marked contrast with epidemics of sexually transmitted diseases or computer viruses to which scale-free network models often apply.