Search

The correlated biased random walk with latency in one and two dimensions is discussed with regard to the portion of irreducible random movement and structured movement. It is shown how a quantitative analysis can be carried out by using computational mechanics. The stochastic matrix for both dynamics are reported. Latency introduces new states in the finite state machine description of the system in both dimensions, allowing for a full nearest neighbor coordination in the two dimensional case. Complexity analysis is used to characterize the movement, independently of the set of control parameters, making it suitable for the discussion of other random walk models. The complexity map of the system dynamics is reported for the two dimensional case.

In the paper, A quantum decomposition (QD, for short) of random walk on Cayley graph of finite group is introduced, which contains two cases. One is QD of quantum random walk operator (QRWO, for short), another is QD of Quantum random walk state (QRWS, for short). Using these findings, I finally obtain some applications for quantum random walk (QRW, for short), which are of interest in the study of QRW, highlighting the role played by QRWO and QRWS.

Exact solutions are rare for non-Markovian random walk models even in 1D, and much more so in 2D. Here we propose a 2D genuinely non-Markovian random walk model with a very rich phase diagram, such that the motion in each dimension can belong to one of 3 categories: (i) subdiffusive, (ii) superdiffusive, or (iii) normally diffusive. The main advance reported here is a different method, and the consequent physical insight, for analytically solving this model. Simpler non-Markovian models, such as Levy walks, have been solved in 2D, but it is not clear if the method of solution could be made to work for more complicated models such as the one studied here. We also report the exact solutions for the first two moments of the random walk propagator, along with the complete phase diagram. The latter is surprisingly rich and admits diverse diffusion regimes. Finally we discuss these results in the context of theoretical underpinnings as well as possible applications.

To obtain an efficient diffusion process is an intriguing and important issue in the study of dynamical behaviors on real networks. Most previous studies are mainly focused on the analysis based on the random walk strategy, whose entropy rate is bounded by the logarithm of the largest node degree of a given graph. In this paper, we take into account a novel strategy named Levy walk and derive the general expression of entropy rate of Levy walk on networks. We present numerical evidences for how the Levy walk strategy delivers an efficient diffusion process on networks and significantly increases the entropy rate compared with the random walk strategy. It is further demonstrated that the capability of Levy walk heavily relies on the network topology as well as the amount of information available regarding the network structure. Specifically, the behavior of Levy walk is highly sensitive to the distribution of shortest distances of the network and its variation. To address this finding, we thereby give a theoretical explanation of the relationship between the variation of shortest distances and the entropy rate of Levy walk. This work may help to enrich our understanding of the behavior of Levy walk and further guide us to find an efficient diffusion process on networks.

The solution of the one-dimensional persistent, biased random walk is found. Its finite differences equation is derived and shown to be satisfied by the said solution.

Conolly et al. [Math. Scientist 22 (1997) 83–91] have obtained the transient distribution for a random walk moving on the integers - ∞ k ∞ of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having “chemical” rule and starting initially at any arbitrary integer position (say i ). As a special case of our result, Conolly et al.'s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated.

Random Boolean networks are among the best-known systems used to model genetic networks. They show an on–off dynamics and it is easy to obtain analytical results with them. Unfortunately very few genes are strictly on–off switched. On the other hand, continuous methods are in principle more suitable to capture the real behavior of the genome, but have difficulties when trying to obtain analytical results. In this work, we introduce a new model of random discrete network: random walk networks, where the state of each gene is changed by small discrete variations, being thus a natural bridge between discrete and continuous models.

We discuss search, minimal path and minimal spanning tree algorithms and their applications to disordered systems. Greedy algorithms solve these problems exactly, and are related to extremal dynamics in physics. Minimal cost path (Dijkstra) and minimal cost spanning tree (Prim) algorithms provide extremal dynamics for a polymer in a random medium (the KPZ universality class) and invasion percolation (without trapping) respectively.

We focus on two models of nearest-neighbour random walks on d-dimensional regular hyper-cubic lattices that are usually assumed to be identical - the discrete-time Polya walk, in which the walker steps at each integer moment of time, and the Montroll-Weiss continuous-time random walk in which the time intervals between successive steps are independent, exponentially and identically distributed random variables with mean 1. We show that while for symmetric random walks both models indeed lead to identical behaviour in the long time limit, when there is an external bias they lead to markedly different behaviour., Comment: 5 pages, 1 figure

An analysis is presented of anti-persistent random walks, in which after any step there is an increased probability of returning to the original site. In addition to their intrinsic interest in the theory of random walks, they are relevant to certain problems in hopping transport, such as ionic conduction. An analysis is presented of such walks on hypercubic lattices in an arbitrary number of dimensions, for both discrete time and continuous time random walks, and exact formulae are derived for the mean square distance travelled in n steps or in time t. The relevance of these results to the observed frequency dependence of ionic conductivity is discussed.

Since the random walk problem was first presented by Pearson in 1905, the shape of a walk which is either completely random or self-avoiding has attracted the attention of generations of researchers working in such diverse fields as chemistry, physics, biology and statistics. Among many advances in the field made in the past decade is the formulation of the three-dimensional shape distribution function of a random walk as a triple Fourier integral plus its numerical evaluation and graphical illustration. However, exact calculations of the averaged individual principal components of the shape tensor for a walk of a certain architectural type including an open walk have remained a challenge. Here we provide an exact analytical approach to the shapes of arbitrary random walks in two dimensions. Especially, we find that an end-looped random walk surprisingly has an even larger shape asymmetry than an open walk.

The authors investigate the random walk of a particle on a one-dimensional chain which has been constructed by a random-walk procedure. Exact expressions are given for the mean-square displacement and the fourth moment after n steps. The probability density after n steps is derived in the saddle-point approximation, for large n. These quantities have also been studied by numerical simulation. The extension to continuous time has been made where the particle jumps according to a Poisson process. The exact solution for the self-correlation function has been obtained in the Fourier and Laplace domain. The resulting frequency-dependent diffusion coefficient and incoherent dynamical structure factor have been discussed. The model of random walk on a random walk is applied to self-diffusion in the concentrated one-dimensional lattice gas where the correct asymptotic behavior is found.

The evolution of a quantum system characterized by a dynamical group G and subjected to random interactions can be reduced, through the use of the generalized coherent states associated with a group G , whose algebra is an ideal of the algebra of G, to a random walk on some homogeneous space S of G . G is the group of automorphisms of S. The connection between the symmetry properties of such a random walk and the structure of G are discussed, showing that it is always possible to map the original process into an equivalent process taking place in the manifold of G. The investigation of the general properties of this mapping quite naturally provides a deep characterization of the role played by G in determining the basic transient and stationary features of the evolution of the system.

We develop the formalism for a continuous-time generalization of the persistent random walk, by allowing the sojourn time to deviate from the exponential form found in standard discussions of this subject. This generalization leads to evolution equations, in the time domain, that differ and are of higher order than the telegrapher's equation which is found in the case of the Markovian persistent random walk.

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.

Complex physical systems are considered, involving both random and deterministic phenomena. We introduce a set of age-dependent transition rates accounting for the random jump processes and a set of deterministic equations describing the evolution of the system between two jumps. The age-state probability density obeys a system of integrodifferential balance equations. Assuming the validity of certain separability conditions, a method is presented for reducing the integration of evolution equations to a discrete random walk problem. For pure jump processes a physical interpretation of the main results is given in terms of the probability density for the waiting time. The approach is applied to heavy ion collisions, to electrodiffusion and to stochastic theory of line shape.

We compare two model descriptions of transport in a material with traps. We establish the equivalence between the n -state trapping model where equilibrium initial conditions are used, and the continuous-time random walk description where a distinct first-waiting-time distribution is introduced. Tunaley's result on the frequency independence of the ac conductivity is corroborated by this equivalence.

The problem of a Polya (unbiased, nearest-neighbor stepping) random walk is solved for several pseudo-lattices related to the Bethe lattice (Cayley tree). In each case, the solution is derived by projecting the walk on a pseudo-lattice onto a walk on a linear chain with internal states and a point defect. Random walk statistics exhibited explicitly include the probability of return to the starting point, the mean time to return if return does occur, and the asymptotic behavior of the expected number of distinct sites visited in a walk of long duration. Conjectured relationships between random walk statistics and percolation theory are discussed in the context of pseudo-lattices.

Ferromagnetic spin systems and gauge theories where the gauge group is topologically a sphere, e.g. Z 2, U(1) and SU(2) are related to the theory of random walk and random surfaces respectively. I survey some applications of this theme to the φ4 field theories.