Persistent random walks in a one-dimensional random environment

Central limit theorems are obtained for persistent random walks in a onedimensional random environment. They also imply the central limit theorem for the motion of a test particle in an infinite equilibrium system of point particles where the free motion of particles is combined with a random collision mechanism and the velocities can take on three possible values.Work supported by the Central Research Fund of the Hungarian Academy of Sciences (grant No. 476/82).     

Asymptotic properties of multistate random walks. II. Applications to inhomogeneous periodic and random lattices

The previously developed formalism for the calculation of asymptotic properties of multistate random walks is used to study random walks on several inhomogeneous periodic lattices, where the periodically repeated unit cell contains a number of inequivalent sites, as well as on lattices with a random distribution of inequivalent sites. We concentrate on the question whether the random walk properties depend on the spatial arrangement of the sites in the unit cell, or only on the number density of the different types of sites. Specifically we consider lattices with periodic and random arrangements of columns and lattices with periodic and random arrangements of anisotropic scatterers.     

First passage time problems for one-dimensional random walks

This note contains a development of the theory of first passage times for one-dimensional lattice random walks with steps to nearest neighbor only. The starting point is a recursion relation for the densities of first passage times from the set of lattice points. When these densities are unrestricted, the formalism allows us to discuss first passage times of continuous time random walks. When they are negative exponential densities we show that the resulting equation is the adjoint of the master equation. This is the lattice analog of a correspondence well known for systems describable by a Fokker-Planck equation. Finally we discuss first passage problems for persistent random walks in which at each step the random walker continues in the same direction as the preceding step with probability a or reverses direction with probability 1–     

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.     

First passage times for correlated random walks and some generalizations

It is generally difficult to solve Fokker-Planck equations in the presence of absorbing boundaries when both spatial and momentum coordinates appear in the boundary conditions. In this note we analyze a simple, exactly solvable model of the correlated random walk and its continuum analogue. It is shown that one can solve for the moments recursively in one dimension in exact analogy with first passage problems for the Fokker-Planck equation, although the boundary conditions are somewhat more complicated. Further generalizations are suggested to multistate random walks.     

A hierarchical model for random walks in random media

We show that the random walk generated by a hierarchical Laplacian in ^d has standard diffusive behavior. Moreover, we show that this behavior is stable under a class of random perturbations that resemble an off-diagonal disordered lattice Laplacian. The density of states and its asymptotic behavior around zero energy are computed: singularities appear in one and two dimensions.     

Scaling relationships for anisotropic random walks

Aspects of transport in a highly multiple-scattering environment are investigated by examining random walkers moving in media having anisotropic angular scattering cross sections (turn-angle distributions). A general expression is obtained for the mean square displacement x² of a random walker executing ann-step walk in an infinite homogeneous material, and results are used to predict scaling relations for the probability() that a walker returns to the planar surface of a semi-infinite medium at a distance from the point of its insertion.     

Fractal random walks

We consider a class of random walks (on lattices and in continuous spaces) having infinite mean-squared displacement per step. The probability distribution functions considered generate fractal self-similar trajectories. The characteristic functions (structure functions) of the walks are nonanalytic functions and satisfy scaling equations.Supported by the Commonwealth Scientific and Industrial Research Organization (Australia).Supported by the Xerox Corporation.Supported in part by a grant from DARPA.     

A study on correlated exponential random walks

Based on the linear Boltzmann transport formulation, we investigate the statistics of correlated exponential random walks that are continuous in space and discrete in time. We show that asymptotically, the correlated random walk process is diffusive and derive an effective diffusion constant. We investigate the power spectral characteristics of the associated random forces. We also present some results on the first passage time distribution and establish that asymptotically it reduces to that associated with simple Gaussian walks.     

Trapping without traps by correlated random walks

Correlated random walk of particles in the infinite cluster of percolating lattices in two dimensions is investigated. For infinitely strong forward correlations (no change of direction except at the boundaries) trapping of the particles in small regions of the infinite cluster is observed.     

Generalized master equations for continuous-time random walks

An equivalence is established between generalized master equations and continuous-time random walks by means of an explicit relationship between(t), which is the pausing time distribution in the theory of continuous-time random walks, and(t), which represents the memory in the kernel of a generalized master equation. The result of Bedeaux, Lakatos-Lindenburg, and Shuler concerning the equivalence of the Markovian master equation and a continuous-time random walk with an exponential distribution for(t) is recovered immediately. Some explicit examples of(t) and(t) are also presented, including one which leads to the equation of telegraphy.This study was partially supported by ARPA and monitored by ONR Contract No. (N00014-17-C-0308).For continuity, the reader is directed to the article entitled Random Walks on Lattices. IV. Continuous Time Walks and Influence of Absorbing Boundaries, by E. W. Montroll and H. Scher, which will appear in Volume 9, Number 2, of this journal, and which should precede the following article. Regrettably, the two articles were inadvertently switched during processing.     

Escape times in interacting biased random walks

The dynamics ofN particles with hard core exclusion performing biased random walks is studied on a one-dimensional lattice with a reflecting wall. The bias is toward the wall and the particles are placed initially on theN sites of the lattice closest to the wall. ForN=1 the leading behavior of the first passage timeT _FP to a distant sitel is known to follow the Kramers escape time formulaT _FP ^l where is the ratio of hopping rates toward and away from the wall. ForN > 1 Monte Carlo and analytical results are presented to show that for the particle closest to the wall, the Kramers formula generalizes toT _FR ^IN. First passage times for the other particles are studied as well. A second question that is studied pertains to survival timesT _s in the presence of an absorbing barrier placed at sitel. In contrast to the first passage time, it is found thatT _s follows the leading behavior independent ofN.     

The Hausdorff dimension of random walks and the correlation length critical exponent in Euclidean field theory

We study the random walk representation of the two-point function in statistical mechanics models near the critical point. Using standard scaling arguments, we show that the critical exponentv describing the vanishing of the physical mass at the critical point is equal tov /d_w, whered _w is the Hausdorff dimension of the walk, andv is the exponent describing the vanishing of the energy per unit length of the walk at the critical point. For the case ofO(N) models, we show thatv ₀=, where is the crossover exponent known in the context of field theory. This implies that the Hausdorff dimension of the walk is/v forO(N) models.     

A note on occupation times of random walks

In this note I show that some asymptotic results on the average occupancy time of an interval derived for lattice random walks with negative exponential transition probabilities are true for all random walks whose transition probabilities have a finite variance. The proof is based on the continuum limit.     

Asymptotic behavior of densities for two-particle annihilating random walks

Consider the system of particles on ^d where particles are of two types—A andB—and execute simple random walks in continuous time. Particles do not interact with their own type, but when anA-particle meets aB-particle, both disappear, i.e., are annihilated. This system serves as a model for the chemical reactionA+B inert. We analyze the limiting behavior of the densities _A (t) and _B (t) when the initial state is given by homogeneous Poisson random fields. We prove that for equal initial densities _A (0)= _B (0) there is a change in behavior fromd4, where _A (t)= _B (t)C/t ^d /4, tod4, where _A (t)= _B (t)C/tast. For unequal initial densities _A (0)< _B (0), _A (t)e ^–cl ind=1, _A (t)e ^–Ct/logt ind=2, and _A (t)e ^–Ct ind3. The termC depends on the initial densities and changes withd. Techniques are from interacting particle systems. The behavior for this two-particle annihilation process has similarities to those for coalescing random walks (A+AA) and annihilating random walks (A+Ainert). The analysis of the present process is made considerably more difficult by the lack of comparison with an attractive particle system.     

Random walks on periodic and random lattices. II. Random walk properties via generating function techniques

We investigate the random walk properties of a class of two-dimensional lattices with two different types of columns and discuss the dependence of the properties on the densities and detailed arrangements of the columns. We show that the row and column components of the mean square displacement are asymptotically independent of the details of the arrangement of columns. We reach the same conclusion for some other random walk properties (return to the origin and number of distinct sites visited) for various periodic arrangements of a given relative density of the two types of columns. We also derive exact asymptotic results for the occupation probabilities of the two types of distinct sites on our lattices which validate the basic conjecture on bond and step ratios made in the preceding paper in this series.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE 78-21460.     

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Lévy distribution with exponent 1/2 as the boundary tends to ∞.     

Asymptotic results on occupancy times of random walks

In this note we derive, using Wald's theorem asymptotic results on mean occupancy time of an interval for random walks with arbitrary transition probabilities. We show that our results are consistent with those obtained (by Weiss, Ref. 2) via the master equation approach, by demonstrating that the resulting infinite series can be summed exactly.     

Correlated random walks

We present a new approach to the calculation of first passage statistics for correlated random walks on one-dimensional discrete systems. The processes may be non-Markovian and also nonstationary. A number of examples are used to demonstrate the theory.