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.     

The dynamics of random interfaces in phase transitions

A review is given of recent developments involving the dynamics of random interfaces formed in the evolution of metastable and unstable systems. Topics which are discussed include interface growth and nonequilibrium dynamical scaling. The possibility of there being dynamical universality classes in first-order phase transitions is also discussed. There are a large number of systems of experimental interest which include binary alloys, binary fluids, and polymer mixtures. Other systems studied by computer simulation include the kinetic Ising, Potts, andZ _N models.Work supported by NSF grant No. DMR-8013700.     

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.     

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.     

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.     

Characteristics of phase transitions via intervention in random networks

We present a percolation process in which the classical Erdo¨s–Re′nyi(ER) random evolutionary network is intervened by the product rule(PR) from some moment t0. The parameter t0is continuously tunable over the real interval [0, 1].This model becomes the random network under the Achlioptas process at t0= 0 and the ER network at t0= 1. For the percolation process at t0≤ 1, we introduce a relatively slow-growing point, after which the largest cluster begins growing faster than that in the ER model. A weakly discontinuous transition is generated in the percolation process at t0≤ 0.5.We take the relatively slow-growing point as the lower pseudotransition point and the maximum gap point of the order parameter as the upper pseudotransition point. The critical point can be approximately predicted by each fitting function of the two points about t0. This contributes to understanding the rapid mergence of the large clusters at the critical point.The numerical simulations indicate that the lower pseudotransition point and the upper pseudotransition point are equal in the thermodynamic limit. When t0> 0.5, the percolation processes generate a continuous transition. The scaling analyses of several quantities are presented, including the relatively slow-growing point, the duration of the relatively slow-growing process, as well as the relatively maximum strength between the percolation percolation at t0< 1 and the ER network about different t0. The presented mechanism can be viewed as a two-stage percolation process that has many potential applications in the growth processes of real networks.     

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.     

Critical disorder and phase transitions in random diode arrays

Random diode arrays represent a new class of nonlinear disordered systems related to the physics of thin-film semiconductor structures and some others. When a disorder strength grows through a certain critical value, they undergo a phase transition from almost uniform to strongly nonuniform random electric potential. A piecewise continuous topography of random potential is predicted.     

Quantum phase transitions in a frustration-free spin chain based on modified Motzkin walks

Area law violations for entanglement entropy in the form of a square root has recently been studied for one-dimensional frustration-free quantum systems based on the Motzkin walks and their variations. Here, we further modify the Motzkin walks using the elements of a symmetric inverse semigroup as basis states on each step of the walk. This change alters the number of paths allowed in the Motzkin walks and by introducing an appropriate term in the Hamiltonian with a tunable parameter we show that we can jump from a state that violates the area law logarithmically to a state that obeys the area law providing an example of quantum phase transition in a one-dimensional system.     

Absence of phase transitions in certain one-dimensional long-range random systems

An Ising chain is considered with a potential of the formJ(i, j)/|i–j|, where theJ(i, j) are independent random variables with mean zero. The chain contains both randomness and frustration, and serves to model a spin glass. A simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if>1. This is to be contrasted with a ferromagnetic interaction which requires>2. The basic idea is to prove that the surfacefree energy between two half-lines is finite, although the surface energy may be unbounded. Ford-dimensional systems, it is shown that the free energy does not depend on the specific boundary conditions if>(1/2)d.     

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.     

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     

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.