Random walk in a random medium in one dimension

Applying scaling and universality arguments, the long-time behavior of the probability distribution for a random walk in a one-dimensional random medium satisfying Sinai's constraint is obtained analytically. The convergence to this asymptotic limit and the fluctuations of this distribution are evaluated by solving numerically the stochastic equations for this walk.     

A generalized Pearson random walk allowing for bias

We calculate asymptotic values of the first two moments of a planar walk in which the step lengths depend on the direction of motion. The model is suggested by experiments on the locomotion of biological cells. Internally induced persistence due to nonuniform turn angle distributions is also accounted for.     

The generalized point interaction in one dimension

We show that various self-adjoint extensions of the operatorH ₀ ⁰ =–d²/dx²,D(H ₀ ⁰ )= =C (R{o}) describes a particle moving under additional influence of the generalized Fermi pseudopotential.I am indebted to Dr. H. Englisch for stimulating discussions. I am also grateful to the Department of Mathematics at the Karl-Marx-Universität Leipzig where this work was done.     

Transitive ensembles of random matrices related to orthogonal polynomials

Transitive correlations of eigenvalues for random matrix ensembles intermediate between real symmetric and hermitian, self-dual quaternion and hermitian, and antisymmetric and hermitian are studied. Expressions for exact n-point correlation functions are obtained for random matrix ensembles related to general orthogonal polynomials. The asymptotic formulas in the limit of large matrix dimension are evaluated at the spectrum edges for the ensembles related to the Legendre polynomials. The results interpolate known asymptotic formulas for random matrix eigenvalues.     

Distribution of roots of random real generalized polynomials

The average density of zeros for monic generalized polynomials, , with real holomorphic ,f _k and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion . The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen.     

A long-time tail for random walk in random scenery

Consider a simple random walk on ^d whose sites are colored black or white independently with probabilityq, resp. 1–q. Walk and coloring are independent. Letn _k be the number of steps by the walk between itskth and (k+1) th visits to a black site (i.e., the length of itskth white run), and let _k =E(n _k )–q ^–1. Our main result is a proof that (*) lim _k k ^d/2 _k = (1 –q)q ^{d/2 – 2}(d/2) ^d/2. Since it is known thatq ^{– 1} _k =E(n ₁ n _{k + 1} B) –E(n ₁ B)E(n _{k + 1} B), withB the event that the origin is black, (*) exhibits a long-time tail in the run length autocorrelation function. Numerical calculations of _k (1k100) ind=1, 2, and 3 show that there is an oscillatory behavior of _k for smallk. This damps exponentially fast, following which the power law sets in fairly rapidly. We prove that if the coloring is not independent, but is convex in the sense of FKG, then the decay of _k cannot be faster than (*).     

Shock profiles for the asymmetric simple exclusion process in one dimension

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at ratesp and 1-p (herep > 1/2) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers’ equation; the latter has shock solutions with a discontinuous jump from left density ρ- to right density ρ+, ρ-< ρ +, which travel with velocity (2p−1 )(1−ρ₊−p ₋). In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time-invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice siten, measured from this particle, approachesp _± at an exponential rate asn→ ±∞, witha characteristic length which becomes independent ofp when . For a special value of the asymmetry, given byp/(1−p)=p ₊(1−p ₋)/p ₋(1−p ₊), the measure is Bernoulli, with densityρ ₋ on the left andp ₊ on the right. In the weakly asymmetric limit, 2p−1 → 0, the microscopic width of the shock diverges as (2p+1)^-1. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.     

A continuous time random walk approach to magnetic disaccommodation

We extend the Dietze theory for the diffusion after-effect to the case where the defects perform a continuous time random walk. Using a waiting time density of the fractional exponential type ψ(t) = (1−n)vt⁻ⁿe^-vt1−n a temporal dependence of a fractional power type t¹⁻ⁿ at short times is reported.     

Dynamical system related to quasiperiodic Schrödinger equations in one dimension

A dynamical map is obtained from a class of quasiperiodic discrete Schrödinger equations in one dimension which include the Fibonacci system. The potentials are constant except for steps at special points.     

A Poisson random walk is Bernoulli

Particles distributed on the integers in Poisson distribution, each independently taking a random walk, form a stationary Markov chain. The canonical shift in this space is Bernoulli.This research was supported in part by NSF grant MCS 78-07739-A03     

From random walk to single-file diffusion

We report an experimental study of diffusion in a quasi-one-dimensional (q1D) colloid suspension which behaves like a Tonks gas. The mean squared displacement as a function of time is described well with an ansatz encompassing a time regime that is both shorter and longer than the mean time between collisions. The ansatz asserts that the inverse mean squared displacement is the sum of the inverse mean squared displacement for short time normal diffusion (random walk) and the inverse mean squared displacement for asymptotic single-file diffusion (SFD). The dependence of the 1D mobility in the SFD on the concentration of the colloids agrees quantitatively with that derived for a hard rod model, which confirms for the first time the validity of the hard rod SFD theory. We also show that a recent SFD theory by Kollmann leads to the hard rod SFD theory for a Tonks gas.     

A restricted random walk on a one-dimensional random chain

It is found that under some special conditions the inverse velocity for a restricted random walk diverges even though it is finite for the corresponding unrestricted walk. This leads to an anomalous t^z (0<z<1) behaviour for the mean distance travelled in time t.     

A random walk representation of the Dirac propagator

We define a discrete random walk with a matrix-valued transition function and show that the scaling limit of the two-point function of the walk is given by the Dirac propagator. We study the scaling limit of similar walks with curvature-dependent transition functions, which are analogous to the Ornstein-Uhlenbeck process, and show that the Dirac propagator can be recovered by a limiting procedure.     

A random walk on a random channel with absorbing barriers

We consider random walks on one-dimensional random channels between two absorbing barriers. The problem can perhaps be used to model the diffusion of a molecule in a “random” membrane, the molecule traversing a random channel formed by the constituent membrane molecules. We are able to analytically follow the transition from diffusive to non-diffusive behavior as the minimum number of channel segments required to traverse the membrane increases.     

A theory of coupled random walk process

A theory of the coupled random walk (CRW) process which has been proposed by one of the present authors, is developed further. It is shown that the Fokker-Planck equation obtained from the CRW process can be cast into a form of a kinetic equation similar to the Boltzmann equation of the gas theory. To this end, the memory effects are taken into account through the jumping probabilities between the modes. As a special case, the distribution function for the steady state is considered.     

A novel identity from random walk theory

A novel expansion of binomial coefficients in terms of trigonometric functions has been obtained by comparing expressions for the time evolution of the probability distribution for a random walker on a ring obtained by separate combinatoric and eigenvalue approaches.     

The generalized plasma in one dimension: Evaluation of a partition function

Forrester and Jancovici have given sum rules for a two-dimensional generalized plasma with two species of particles interacting through logarithmic potentials with three independent coupling constants. They have also found a specific one-dimensional solvable model which satisfies the analogs of their sum rules. A class of one-dimensional models for which the partition function is evaluable is given as well as a more general result evaluating multi-dimensional integrals.Partially supported by N.S.F. grant     

Eigenfunction approach to the persistent random walk in two dimensions

The Fourier–Bessel expansion of a function on a circular disc yields a simple series representation for the end-to-end probability distribution function w(R,φ) encountered in a planar persistent random walk, where the direction taken in a step depends on the relative orientation towards the preceding step. For all but the shortest walks, the proposed method provides a rapidly converging, numerically stable algorithm that is particularly useful for the precise study of intermediate-size chains that have not yet approached the diffusion limit.a