Random walks on pseudo-lattices

The problem of a Pólya (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.     

Pólya number and first return of bursty random walk: Rigorous solutions

The recurrence properties of random walks can be characterized by Pólya number, i.e., the probability that the walker has returned to the origin at least once. In this paper, we investigate Pólya number and first return for bursty random walk on a line, in which the walk has different step size and moving probabilities. Using the concept of the Catalan number, we obtain exact results for first return probability, the average first return time and Pólya number for the first time. We show that Pólya number displays two different functional behavior when the walk deviates from the recurrent point. By utilizing the Lagrange inversion formula, we interpret our findings by transferring Pólya number to the closed-form solutions of an inverse function. We also calculate Pólya number using another approach, which corroborates our results and conclusions. Finally, we consider the recurrence properties and Pólya number of two variations of the bursty random walk model.     

Scaling Argument of Anisotropic Random Walk

In this paper, we analytically discuss the scaling properties of the average square end-to-end distance 〈R²〉for anisotropic random walk in D-dimensional space (D≥2), and the returning probability P_n( r₀) for the walker into a certain neighborhood of the origin. We will not only give the calculating formula for 〈R²〉and P_n(r₀), but also point out that if there is a symmetric axis for the distribution of the probability density of a single step displacement, we always obtain 〈R²_⊥n〉～n, where ⊥ refers to the projections of the displacement perpendicular to each symmetric axes of the walk; in D-dimensional space with D symmetric axes perpendicular to each other, we always have 〈R_n²〉～n and the random walk will be like a purely random motion; if the number of inter-perpendicular symmetric axis is smaller than the dimensions of the space, we must have 〈R_n²〉～n² for very large n and the walk will be like a ballistic motion. It is worth while to point out that unlike the isotropic random walk in one and two dimensions, which is certain to return into the neighborhood of the origin, generally there is only a nonzero probability for the anisotropic random walker in two dimensions to return to the neighborhood.     

Random walk on a one-dimensional lattice with a random distribution of traps

The probability of return to the starting point of a particle executing a random walk on a one-dimensional lattice with a static distribution of traps, is derived, for asymptotically large n, the number of steps. The probability decreases as exp ($\text{?n}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}$), in agreement with some recent estimates of its upper and lower bounds for diffusion in d dimensions.     

Asymptotic properties of multistate random walks. I. Theory

A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) formultistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of a periodically repeated unit cell which contains a finite number of internal states (sites). The results are identical to those for perfect lattices except for a renormalization of coefficients. For walks without drift, it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk. The diffusion coefficients can be obtained by a simple matrix algorithm presented here. Both discrete and continuous time random walks are considered. The results are not restricted to nearest-neighbor random walks but apply as long as the single-step probability distributions associated with each of the internal states have finite means and variances.     

Biased random walk on a multidimensional lattice with absorbing boundaries

This paper deals with continuous-time random walk on a hypercubic lattice with absorbing boundaries along the axial planes through the origin. Possible bias in the transition probabilities along any axis is allowed for. Any dynamical model which is solvable in the case of an infinite lattice is shown to be tractable also in the present case. As an example, the exponential holding time model is solved explicitly. Representative numerical results for the probability of return to the starting point and the probability of absorption are presented for the one-dimensional case. It is found that, unlike the absorption which increases with bias towards the boundary, the return probability is independent of the direction of the bias.     

Return Probability of the Open Quantum Random Walk with Time-Dependence

We study the open quantum random walk (OQRW) with time-dependence on the one-dimensional lattice space and obtain the associated limit distribution. As an application we study the return probability of the OQRW. We also ask, "What is the average time for the return probability of the OQRW?"     

Mechanisms and energetic of atomic processes in surface diffusion

Surface diffusion is a subject of basic importance for understanding mass transport phenomena in surface and nano science. In the particle aspect of surface diffusion of single atoms and simple molecules, information of interest is the detail atomic mechanisms and the activation energy of various atomic processes, and also the binding energy of atoms at different surface sites. In the absence of an external force, atoms will perform random walk without a preferred direction. When an atom is subjected to an external force, or when a chemical potential gradient exists, it will move preferentially in the direction of the force, or in the direction of decreasing chemical potential, thus the random walk becomes directional. Using atomic resolution microscopy, it is now possible to observe random walk diffusion of atoms, molecules and atomic clusters directly as well as to study the dynamic behavior of atoms as perturbed by the electronic interactions of the surface in great detail. Here, methods of studying quantitatively the particle aspect of surface diffusion and how it affects the dynamic behavior of the surface are very briefly reviewed.     

The two-state random walk

We develop asymptotic results for the two-state random walk, which can be regarded as a generalization of the continuous-time random walk. The two-state random walk is one in which a particle can be in one of two states for random periods of time, each of the states having different spatial transition probabilities. When the sojourn times in each of the states and the second moments of transition probabilities are finite, the state probabilities have an asymptotic Gaussian form. Several known asymptotic results are reproduced, such as the Gaussian form for the probability density of position in continuous-time random walks, the time spent in one of these states, and the diffusion constant of a two-state diffusing particle.     

Random walk on a random 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.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

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.     

Microscopic Approach to Random Walks

In the present paper the microscopic approach to random walk models is introduced. For any particular model it provides a rigorous way to derive the transport equations for the macroscopic density of walking particles. Although it is not more complicated than the standard random walk framework it has virtually no limitations with respect to the initial distribution of particles. As a consequence, the transport equations derived with this method almost automatically give answers to such important problems as aging and two point probability distribution.     

Derivation of the Boltzmann equation for a Fermi gas

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant ^?3/2.     

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.     

A Note on Transience Versus Recurrence for a Branching Random Walk in Random Environment

We consider a branching random walk in random environment on ^d where particles perform independent simple random walks and branch, according to a given offspring distribution, at a random subset of sites whose density tends to zero at infinity. Given that initially one particle starts at the origin, we identify the critical rate of decay of the density of the branching sites separating transience from recurrence, i.e., the progeny hits the origin with probability <1 resp. =1. We show that for d3 there is a dichotomy in the critical rate of decay, depending on whether the mean offspring at a branching site is above or below a certain value related to the return probability of the simple random walk. The dichotomy marks a transition from local to global behavior in the progeny that hits the origin. We also consider the situation where the branching sites occur in two or more types, with different offspring distributions, and show that the classification is more subtle due to a possible interplay between the types. This note is part of a series of papers by the second author and various co-authors investigating the problem of transience versus recurrence for random motions in random media.     

Random walks in the space of conformations of toy proteins

Monte Carlo dynamics of the lattice toy protein of 48 monomers is interpreted as a random walk in an abstract (discrete) space of conformations. To test the geometry of this space, we examine the return probability P(T), which is the probability to find the polymer in the native state after T Monte Carlo steps, provided that it starts from the native state at the initial moment. Comparing computational data with the theoretical expressions for P(T) for random walks in a variety of different spaces, we show that conformation spaces of polymer loops may have nontrivial dimensions and exhibit negative curvature characteristics of Lobachevskii (hyperbolic) geometry.     

One-dimensional stochastic Levy-lorentz gas

We introduce a Levy-Lorentz gas in which a light particle is scattered by static point scatterers arranged on a line. We investigate the case where the intervals between scatterers xi(i) are independent random variables identically distributed according to the probability density function &mgr;(xi) approximately xi(-(1+gamma)). We show that under certain conditions the mean square displacement of the particle obeys /=Ct3-gamma for 1相似文献   

初态对光波导阵列中连续量子行走影响的研究

本文使用近邻耦合模型得到的解析解,分析了周期性波导中输入态对量子行走的粒子数的概率分布函数 和二阶相干性的影响.结果表明:输入态的对称性质对量子行走过程的二阶相干度有影响, 而对粒子数的概率分布函数影响不大. 关键词： 周期性光波导阵列 量子行走 二阶相干度 纠缠态     

Random walks on lattices with randomly distributed traps I. The average number of steps until trapping

For a random walk on a lattice with a random distribution of traps we derive an asymptotic expansion valid for smallq for the average number of steps until trapping, whereq is the probability that a lattice point is a trap. We study the case of perfect traps (where the walk comes to an end) and the extension obtained by letting the traps be imperfect (i.e., by giving the walker a finite probability to remain free when stepping on a trap). Several classes of random walks of varying dimensionality are considered and special care is taken to show that the expansion derived is exact up to and including the last term calculated. The numerical accuracy of the expansion is discussed.