Are random walks random?

Random walks have been created using the pseudo-random generators in different computer language compilers (BASIC, PASCAL, FORTRAN, C++) using a Pentium processor. All the obtained paths have apparently a random behavior for short walks (2¹⁴ steps). From long random walks (2³³ steps) different periods have been found, the shortest being 2¹⁸ for PASCAL and the longest 2³¹ for FORTRAN and C++, while BASIC had a 2²⁴ steps period. The BASIC, PASCAL and FORTRAN long walks had even (2 or 4) symmetries. The C++ walk systematically roams away from the origin. Using deviations from the mean-distance rule for random walks, d² ∝ N, a more severe criterion is found, e.g. random walks generated by a PASCAL compiler fulfills this criterion to N < 10 000.     

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.     

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.     

Directed random walks in random environments

We use holding time methods to study the asymptotic behavior of pure birth processes with random transition rates. Both the normal and slow approaches to infinity are studied. Fluctuations are shown to obey the central limit theorem for almost all sample-transition rates. Our results are stronger, and our proofs simpler, then those of recently published studies.     

Symmetric random walks in random environments

We consider a random walk on thed-dimensional lattice ^d where the transition probabilitiesp(x,y) are symmetric,p(x,y)=p(y,x), different from zero only ify–x belongs to a finite symmetric set including the origin and are random. We prove the convergence of the finite-dimensional probability distributions of normalized random paths to the finite-dimensional probability distributions of a Wiener process and find our an explicit expression for the diffusion matrix.     

The fractal dimension in computer-simulated random walks

Computer-simulated random walks of Powles and Rapaport have been analyzed in terms of a scale-dependent fractal dimension D(r). It turns out that these random walks perfectly fit this D-expression. This suggests that a theoretical derivation of this D-form should be possible in three-dimensional systems.     

Short linear random walks

Linear random walks are usually described by formulae valid for a large number of steps. Formulae applicable to short walks and to walks in the presence of traps as well as of rotations are given and the pertinent activation energies in thermally activated processes are discussed. Applicability to interpretation of experimental data is pointed out.     

The critical behaviour of two-dimensional self-avoiding random walks

We present exact results for the mean end-to-end distance of self-avoiding random walks on several planar lattices. For the square lattice, we extend the known results from walks with 20 steps to walks with 22 steps, and for the triagular lattice from 14 to 16 steps. For the honeycomb lattice we went up to 34 steps, for the two-choice square lattice up to 44 steps, and for the 4-choice triagular lattice up to 19 steps. The extrapolated valuev=0.747±0.001 (provided the correction-to-scalng exponent is not appreaciably smaller than unity) is in disagreement with both Flory's value and the recent estimate of Derrida. We claim that a different analysis of Derrida's data supports this value.Address from 1st April–30th September 1982: Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, IsraelAddress from 1st April-30th September 1982, Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel     

Localization of random walks in one-dimensional random environments

We consider a random walk on the one-dimensional semi-lattice ={0, 1, 2,...}. We prove that the moving particle walks mainly in a finite neighbourhood of a point depending only on time and a realization of the random environment. The size of this neighbourhood is estimated. The limit parameters of the walks are also determined.     

Deterministic walks in random environments

Deterministic walks in random environments (DWRE) occupy an intermediate position between purely random (generated by random trials) and purely deterministic (generated by deterministic dynamical systems, e.g., by maps) models of diffusion. These models combine deterministic and probabilistic features. We review general properties of DWRE and demonstrate that, to a large extent, their dynamics and their statistics can be analyzed consecutively and separately. We also show that orbits of one-dimensional walks in rigid environments with non-constant rigidity almost surely visit each site infinitely many times.     

Discrete versus continuous-time random walks

The results obtained on the basis of discrete and continuous-time random walk models on a finite chain are compared with one another in problems such as longitudinal dispersion and the spectrum of a random oscillator. In these applications, discrete and continuous-time models cannot be used inter-changeably.     

From random to self-avoiding walks

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π _i=0 ^N?2 Π_j=i+2/N(1?ωδ_ij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.     

Self-avoiding walks on random lattices

The properties of self-avoiding walks on dilute lattices are studied, both directly and using the replica formalism. It is shown that dilution does not affect the exponents and careful use of the Haris criterion also leads to this conclusion.     

Loops in one-dimensional random walks

Distribution of loops in a one-dimensional random walk (RW), or, equivalently, neutral segments in a sequence of positive and negative charges is important for understanding the low energy states of randomly charged polymers. We investigate numerically and analytically loops in several types of RWs, including RWs with continuous step-length distribution. We show that for long walks the probability density of the longest loop becomes independent of the details of the walks and definition of the loops. We investigate crossovers and convergence of probability densities to the limiting behavior, and obtain some of the analytical properties of the universal probability density. Received 8 January 1999     

The theory of ordered spans of unrestricted random walks

The spans of ann-step random walk on a simple cubic lattice are the sides of the smallest rectangular box, with sides parallel to the coordinate axes, that contains the random walk. Daniels first developed the theory in outline and derived results for the simple random walk on a line. We show that the development of a more general asymptotic theory is facilitated by introducing the spectral representation of step probabilities. This allows us to consider the probability density for spans of random walks in which all moments of single steps may be infinite. The theory can also be extended to continuous-time random walks. We also show that the use of Abelian summation simplifies calculation of the moments. In particular we derive expressions for the span distributions of random walks (in one dimension) with single step transition probabilities of the formP(j) 1/j ¹⁺, where 0<<2. We also derive results for continuous-time random walks in which the expected time between steps may be infinite.     

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).