Modes of a Gaussian Random Walk

It is demonstrated that a one-dimensional gaussian random walk (GRW) possesses an underlying structure in the form of random oscillatory modes. These modes are not sinusoids, but can be isolated by a well-defined procedure. They have average wavelengths and amplitudes, both of which can be determined by experiments or by theoretical calculations. This paper reports such determinations by both methods and also develops a theory that is ultimately shown to agree with experiments. Both theory and simulations show that the average wavelength and the average amplitude scale with the order of the mode in exactly the same way that the modes of the well-known Weierstrass fractal scale with mode order. This is remarkable since the wave generated by the Weierstrass function, , is fully determined for the variable x whereas the GRW is stochastic. By increasing the size of the steps in the GRW, it is possible to selectively remove the fastest modes, while leaving the remaining modes almost unchanged. For a GRW, the parameters corresponding to a and g in the Weierstrass function are found to be 2.0 and 4.0, respectively. These values are independent of the variance associated with the GRW. Application of the random modes is reserved for a later paper.     

Stationary Probability and First-Passage Time of Biased Random Walk

In this paper, we consider the stationary probability and first-passage time of biased random walk on 1D chain, where at each step the walker moves to the left and right with probabilities p and q respectively (0≤p, q≤1, p+q=1). We derive exact analytical results for the stationary probability and first-passage time as a function of p and q for the first time. Our results suggest that the first-passage time shows a double power-law F~(N-1)^γ, where the exponent γ=2 for N<|p-q|^-1 and γ=1 for N>|p-q|^-1. Our study sheds useful insights into the biased random-walk process.     

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.     

Numerical Simulations of Random Walk in Random Environment

This paper is concerned with the numerical simulation of a random walk in a random environment in dimension d = 2. Consider a nearest neighbor random walk on the 2-dimensional integer lattice. The transition probabilities at each site are assumed to be themselves random variables, but fixed for all time. This is the random environment. Consider a parallel strip of radius R centered on an axis through the origin. Let X _R be the probability that the walk that started at the origin exits the strip through one of the boundary lines. Then X _R is a random variable, depending on the environment. In dimension d = 1, the variable X _R converges in distribution to the Bernoulli variable, X = 0, 1 with equal probability, as R . Here the 2-dimensional problem is studied using Gauss-Seidel and multigrid algorithms.     

Random Walk on the Range of Random Walk

We study the random walk X on the range of a simple random walk on ℤ ^d in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.     

Non-Hermitian Tridiagonal Random Matrices and Returns to the Origin of a Random Walk

We study a class of tridiagonal matrix models, the q-roots of unity models, which includes the sign (q=2) and the clock (q=) models by Feinberg and Zee. We find that the eigenvalue densities are bounded by and have the symmetries of the regular polygon with 2q sides, in the complex plane. Furthermore, the averaged traces of M ^k are integers that count closed random walks on the line such that each site is visited a number of times multiple of q. We obtain an explicit evaluation for them.     

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.     

Velocity and Dispersion for a Two-Dimensional Random Walk

In the paper, we consider the transport of a two-dimensional random walk. The velocity and the dispersion of this two-dimensional random walk are derived. It mainly show that： （i） by controlling the values of the transition rates, the direction of the random walk can be reversed; （ii） for some suitably selected transition rates, our two-dimensional random walk can be efficient in comparison with the one-dimensional random walk. Our work is motivated in part by the challenge to explain the unidirectional transport of motor proteins. When the motor proteins move at the turn points of their tracks （i.e., the cytoskeleton filaments and the DNA molecular tubes）, some of our results in this paper can be used to deal with the problem.     

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.     

Wavefunction Collapse and Random Walk

Wavefunction collapse models modify Schrödinger's equation so that it describes the rapid evolution of a superposition of macroscopically distinguishable states to one of them. This provides a phenomenological basis for a physical resolution to the so-called measurement problem. Such models have experimentally testable differences from standard quantum theory. The most well developed such model at present is the Continuous Spontaneous Localization (CSL) model in which a universal fluctuating classical field interacts with particles to cause collapse. One side effect of this interaction is that the field imparts energy to the particles: experimental evidence on this has led to restrictions on the parameters of the model, suggesting that the coupling of the classical field to the particles must be mass-proportional. Another side effect is that the field imparts momentum to particles, causing a small blob of matter to undergo random walk. Here we explore this in order to supply predictions which could be experimentally tested. We examine the translational diffusion of a sphere and a disc, and the rotational diffusion of a disc, according to CSL. For example, we find that the rms distance an isolated 10^–5 cm radius sphere diffuses is (its diameter, 5 cm) in (20 sec, a day), and that a disc of radius 2 10^–5 cm and thickness 0.5 10^–5 cm diffuses through 2rad in about 70 sec (this assumes the standard CSL parameter values). The comparable rms diffusions of standard quantum theory are smaller than these by a factor 10^–3±1. It is shown that the CSL diffusion in air at STP is much reduced and, indeed, is swamped by the ordinary Brownian motion. It is also shown that the sphere's diffusion in a thermal radiation bath at room temperature is comparable to the CSL diffusion, but is utterly negligible at liquid He temperature. Thus, in order to observe CSL diffusion, the pressure and temperature must be low. At the low reported pressure of 5 10^–17 Torr, achieved at 4.2°K, the mean time between air molecule collisions with the (sphere, disc) is (80, 45)min. This is ample time for observation of the putative CSL diffusion with the standard parameters and, it is pointed out, with any parameters in the range over which the theory may be considered viable. This encourages consideration of how such an experiment may actually be performed, and the paper closes with some thoughts on this subject.     

Continuous-Time Classical and Quantum Random Walk on Direct Product of Cayley Graphs

In this paper we define direct product of graphs and give a recipe for obtaining probability of observing particle on vertices in the continuous-time classical and quantum random walk. In the recipe, the probability of observing particle on direct product of graph is obtained by multiplication of probability on the corresponding to sub-graphs, where this method is useful to determining probability of walk on compficated graphs. Using this method, we calculate the probability of Continuous-time classical and quantum random walks on many of finite direct product Cayley graphs （complete cycle, complete Kn, charter and n-cube）. Also, we inquire that the classical state the stationary uniform distribution is reached as t→∞ but for quantum state is not always satisfied.     

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?"     

A General Random Walk Model of Molecular Motor

A general random walk model framework is presented which can be used to statistically describe the internaldynamics and external mechanical movement of molecular motors along filament track. The motion of molecular motorin a periodic potential and a constant force is considered. We show that the molecular motor‘s movement becomesslower with the potential barrier increasing, but if the forceis increased, the molecular motor‘s movement becomesfaster. The relation between the effective rate constant and the potential barrier‘s height, and that between the effectiverate constant and the value of the force are discussed. Our results are consistent with the experiments and relevanttheoretical consideration, and can be used to explain some physiological phenomena.     

A General Random Walk Model of Molecular Motor

A general random walk model framework is presented which can be used to statistically describe the internal dynamics and external mechanical movement of molecular motors along filament track. The motion of molecular motor in a periodic potential and a constant force is considered. We show that the molecular motor‘s movement becomes slower with the potential barrier increasing, but if the force is increased, the molecular motor‘‘s movement becomes faster. The relation between the effective rate constant and the potential battler‘s height, and that between the effective rate constant and the value of the force are discussed. Our results are consistent with the experiments and relevant theoretical consideration, and can be used to explain some physiological phenomena.     

Escape of a Uniform Random Walk from an Interval

We study the first-passage properties of a random walk in the unit interval in which the length of a single step is uniformly distributed over the finite range [−a,a]. For a of the order of one, the exit probabilities to each edge of the interval and the exit time from the interval exhibit anomalous properties stemming from the change in the minimum number of steps to escape the interval as a function of the starting point. As a decreases, first-passage properties approach those of continuum diffusion, but non-diffusive effects remain because of residual discreteness effects. PACS: 02.50.C2, 05.40.Fb     

General Asymptotics of the Density of a Restricted Coalescing Random Walk System

These results explore the asymptotic behavior of the density of a system of coalescing random walks where particles begin from only a subspace of the integer lattice and are allowed to walk anywhere on the lattice. They generalize results by Bramson and Griffeath from 1980.⁽¹⁾ Since the probability that a given site is occupied depends on how far that site is from the originating subspace, the density of the system at a given time must be re-defined. However, the general idea is still that if the density is larger than we expect at a given time, more coalescing events will occur, and the density will correct itself over time.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

Return Probability of the Fibonacci Quantum Walk

In this paper the return probability of the one-dimensional discrete-time quantum walk is studied. We derive probabilistic formulas for the return probability related to the quantum walk governed by the Fibonacci coin.     

On Origin of Power-Law Distributions in Self-Organized Criticality from Random Walk Treatment

The origin of power-law distributions in self-organized criticality is investigated by treating the variation of the number of active sites in the system as a stochastic process. An avalanche is then regarded as a first-return random walk process in a one-dimensional lattice. We assume that the variation of the number of active sites has three possibilities in each update： to increase by 1 with probability f1, to decrease by 1 with probability f2, or remain unchanged with probability 1 - f1 - f2. This mimics the dynamics in the system. Power-law distributions of the lifetime are found when the random walk is unbiased with equal probability to move in opposite directions. This shows that power-law distributions in self-organized criticality may be caused by the balance of competitive interactions.