Random Walk Problem Research Articles

Ladder heights, Gaussian random walks and the Riemann zeta function

Let $\{S_n: n \geq 0\}$ be a random walk having normally distributed increments with mean $\theta$ and variance 1, and let $\tau$ be the time at which the random walk first takes a positive value, so that $S_{\tau}$ is the first ladder height. Then the expected value $E_{\theta} S_{\tau}$, originally defined for positive $\theta$, maybe extended to be an analytic function of the complex variable $\theta$ throughout the entire complex plane, with the exception of certain branch point sin-gularities. In particular, the coefficients in a Taylor expansion about $\theta = 0$ may be written explicitly as simple expressions involving the Riemann zeta function. Previously only the first coefficient of the series developed here was known; this term has been used extensively in developing approximations for boundary crossing problems for Gaussian random walks. Knowledge of the complete series makes more refined results possible; we apply it to derive asymptotics for boundary crossing probabilities and the limiting expected overshoot.

Relaxation in a confined and entangled polymer liquid

We investigate the dynamics of a monodisperse polymer melt that is confined to the vicinity of a plane by a harmonic potential. Polymer molecules are represented by harmonic bead-spring chains with fluctuating bead mobility tensors. Mobility fluctuation rates are determined self-consistently from the chain dynamics. The calculation of the viscoelastic shear modulus is mapped onto the solution of a dynamically disordered random walk, in which a walker hops among lattice sites with fluctuating hopping rates. The relevant random walk problems are solved within the effective medium approximation, yielding the shear modulus and coefficient of shear viscosity as functions of chain length, entanglement molecular weight, and strength of the confining potential. Calculations illustrate the competition between entanglement effects, which retard chain dynamics, and the effect of the external field, which promotes relaxation.

BRANCH ANNIHILATION RANDOM WALKS IN ENERGETIC DISORDERED STRUCTURE AND DIRECTED POLYMER

The problem of the branch annihilation random walks (BAW) in energetic disordered structure is introduced and studied on the checkerboard lattice of one spatial and one temporal dimensions. The energy disorder we considered is the one used in the problem of directed polymer (DPO) at zero temperature including the external field F. The evolution rule of BAW is assigned to relate to the energy disorder. It is found that depending on the value of F, a variety of phases appear, and the crossover from the DPO state to the directed percolation (DPE) state occurs. When F > 0, the problem is reduced to the DPO problem at zero temperature. However, for F < 0, there exist three states, the inactive state for F c < F < 0, the active state for − 1 < F < F c , and the non-hopping state for F < -1, where F c is the threshold value from the inactive state to the active state. The phase transition at F c turns out to be in the DPE universality class. The robustness of the DPE universality is discussed by the Harris criterion.

On some boundary crossing problems for Gaussian random walks

We consider random walks with Gaussian distribution of summands. New representations for Wiener-Hopf factorization components are obtained. The factorization method is used to study the distribution of the excess over one-sided and two-sided boundaries. Asymptotic expansions for these distributions and for the expectation of the first exit time are obtained under the assumption that the boundaries tend to infinity.

Random walk with a boundary line as a free massive boson with a defect line

We show that the problem of Random Walk with boundary attractive potential may be mapped onto the free massive bosonic Quantum Field Theory with a line of defect. This mapping permits to recover the statistical properties of the Random Walks by using boundary $S$--matrix and Form Factor techniques.

Random walks on bundled structures.

Bundled structures (BS) are discrete structures obtained joining to each point of a ``base'' graph a copy of a ``fiber'' graph. In condensed matter physics BS are used as realistic models for the geometry and dynamics of nontranslationally invariant systems (polymers, inhomogeneous systems, etc.). We present an analytical solution for the random walk problem on these structures, which is possible when we know the solution for base and fiber separately. We obtain an expression for the spectral dimension of the BS as a function of the spectral dimensions of its components. Moreover, we discuss some applications of these results concerning anomalous diffusion laws, proving the existence of nondisordered structures with logarithmic and sublogarithmic diffusion laws due only to geometric features.

Ga self-diffusion in GaAs isotope heterostructures.

Isotopically controlled GaAs heterostructures have been used to study Ga self-diffusion with secondary-ion mass spectrometry. This approach probes a close to ideal random walk problem, free from perturbations such as electric fields, mechanical stresses, or chemical potentials. The Ga self-diffusion coefficient in intrinsic GaAs can be well described with $D\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}(43\ifmmode\pm\else\textpm\fi{}25)\mathrm{exp}[(\ensuremath{-}4.24\ifmmode\pm\else\textpm\fi{}0.06\mathrm{eV}){/k}_{B}T]$ over 6 orders of magnitude between 800 and 1225 \ifmmode^\circ\else\textdegree\fi{}C under As-rich condition. No significant doping effects are observed in samples with their substrates doped with Te up to $4\ifmmode\times\else\texttimes\fi{}{10}^{17}{\mathrm{cm}}^{\ensuremath{-}3}$ or Zn up to $1\ifmmode\times\else\texttimes\fi{}{10}^{19}{\mathrm{cm}}^{\ensuremath{-}3}$. Our results substantiate some of the findings in recent theoretical work.

Gamma caméra à collimation assistée par ordinateur (CACAO)Deuxième partie: une nouvelle approche dans l'étude des performances en tomographie d'émission

Computer assisted collimation gamma camera (CACAO in French) is a project to improve both the resolution and sensitivity of single photon emission computerized tomography (SPECT). The principle of CACAO is to use collimators with bigger holes, add some acquisition movement and let the computer achieve the remaining collimation during the tomography treatment. The main theoretical problem here, is the evaluation of the signal to noise ratio and its link with the resolution. To solve this problem, we use information theory, where the data are described in a histogram mode. We are then faced to a random walk problem, limited by only two factors. The first one is the uncertainty over the number of counts in an element of the detector, which clearly depends on the efficiency of the system. The second, which has rarely been taken into account in the literature is the uncertainity over the multiplex volume, which is the volume of the object irradiating the element of detector. The CACAO method could increase both these factors, compared with standard parallel hole collimator SPECT. This will lead to a dramatic increase in the quality of the scintigram. The method described could be useful to optimize the parameter involved in all the tomography and multiplex systems. Further refinement such as Monte Carlo calculation could easily be derived. L'étude des performances (rapport signal sur bruit et résolution) en tomogaphie d'émission gamma, est un problème délicat Nous présentons une approche originale basée sur la théorie de l'information, qui présente par rapport aux techniques existantes l'intérêt d'être indépendante du traitement de reconstruction et de pouvoir s'appliquer au concept de caméra à collimation assistée par ordinateur (CACAO) décrit dans le premier article. L'entropie associée à une mesure élémentaire est calculée en se limitant à un ensemble d'états d'information finis et dénombrables. Pour ce faire, nous travaillons dans l'ensemble des images dont la résolution et la précision statistique sont bornées par un nombre de pixels et un nombre de photons donnés. Des diagrammes de marche au hasard permettent une vue d'ensemble du problème et font apparaître les deux principaux facteurs limitant les performances de la tomoscintigraphie gamma. Ces facteurs sont les incertitudes sur le nombre de photons émis et sur le volume multiplex (volume en regard d'un élément de détecteur). Nous utilisons ensulte ces facteurs pour comparer les tomoscintigraphies pratiquées avec un matériel classique et celles qui devraient être obtenues en utilisant le système CACAO. Nous espérons démontrer les améliorations potentielles du rapport signal sur bruit et de la résolution.

The Grain Boundary Diffusion Kinetics in Nanocrystalline Zirconia

ABSTRACTIt is expected that the grain boundary diffusion is the principal contributor to the transport properties of nanocrystalline zirconia, and that it controls the oxidation kinetics and hydrogen permeation. This process depends on the grain boundary character distribution (i.e. the fractions of different grain boundary types) and on the topological characteristics of the grain boundary network. Modeling of the random walk problem on a planar honeycomb network for different types of the grain boundary misorientation distributions (GBMD) in nanocrystalline zirconia film is presented in the current paper. The GBMD was calculated using the model texture. Changes of the oxidation rate for different types of the grain boundary character distribution and different ratios of the bulk and grain boundary diffusion coefficients are analyzed. In this study it was found that an increase of the frequency of low energy boundaries lowers the oxidation rate.

Function discontinuities in maximum spectral energy probability density function of pseudorandom sequences

The probability density function (PDF) of normalised maximum spectral energy of pseudorandom binary sequences {1, –1} is approximated empirically as a prologue for the statistical connection of spectral flatness to sequence correlation properties. The startling appearance of apparently anomalous δ function discontinuities in the resulting PDF is explained in terms of the random walk problem, and suggests possible distinct properties of sequences linked to those discontinuities.

Toeplitz matrices and random walks with memory

We use a technique based on Toeplitz matrices to calculate the probability distribution for certain random walks on a lattice in continuous time where the walker can take steps of various sizes in each direction and where the probability of a step depends on the nature of a finite set of previous steps. If k( ij) is the rate constant for a step of j units given a history of type i, then we can solve the random walk problem for the special case when the sum over k( ij) is independent of j.

Entropy barriers and slow relaxation in some random walk models

We study the zero temperature limit of a simple model of slow relaxation without energy barriers, proposed by Ritort (1995), as well as two other closely related models with a much faster relaxation. These models can be mapped onto random walk problems, which allows for their analytic study. We analyse, in particular, a specific aspect of the former model, namely the existence of a bias leading to `entropy barriers` and to a very slow relaxation.

Exact shapes of random walks in two dimensions

Since the random walk problem was first presented by Pearson in 1905, the shape of a walk which is either completely random or self-avoiding has attracted the attention of generations of researchers working in such diverse fields as chemistry, physics, biology and statistics. Among many advances in the field made in the past decade is the formulation of the three-dimensional shape distribution function of a random walk as a triple Fourier integral plus its numerical evaluation and graphical illustration. However, exact calculations of the averaged individual principal components of the shape tensor for a walk of a certain architectural type including an open walk have remained a challenge. Here we provide an exact analytical approach to the shapes of arbitrary random walks in two dimensions. Especially, we find that an end-looped random walk surprisingly has an even larger shape asymmetry than an open walk.

An optimal stopping problem for random walks with non-zero drift

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ES τ = ES 1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

Random walks in a quarter plane with zero drifts: transience and recurrence

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.

An optimal stopping problem for random walks with non-zero drift

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ESτ = ES1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

Random walks in a quarter plane with zero drifts: transience and recurrence

In this paper we continue the study of the classification problem for random walks in the quarter plane, with zero drifts in the interior of the domain. The necessary and sufficient conditions for these random walks to be ergodic were found earlier in [3]. Here we obtain necessary and sufficient conditions for transience by constructing suitable Lyapounov functions.

Hyperbolic type transport equations

In recent years hyperbolic type transport equations have acquired a great deal of importance in problems ranging from theoretical physics to biology. In spite of their greater mathematical difficulty as compared with their parabolic type analogs arising from the framework of Linear Irreversible Thermodynamics, they have, in many ways, superseded the latter ones. Although the use of this type of equations is well known since the last century through the telegraphist equation of electromagnetic theory, their use in studying several problems in transport theory is hardly fifty years old. In fact the first appearance of a hyperbolic type transport equation for the problem of heat conduction dates back to Cattaneos' work in 1948. Three years later, in 1951 S. Goldstein showed how in the theory of stochastic processes this type of an equation is obtained in the continuous limit of a one-dimensional persistent random walk problem. After that, other phenomenological derivations have been offered for such equations. The main purpose of this paper is to critically discuss a derivation of a hyperbolic type Fokker-Planck equation recently presented using the same ideas as M.S. Green did in 1952 to provide the stochastic foundations of irreversible statistical mechanics. Arguments are given to show that such an equation as well as transport equations derived from it by taking appropriate averages are at most approximate and that a much more detailed analysis is required before asserting their validity.

Dynamic structure factor in a bidisperse polymer melt

We present calculations of the single-chain dynamic structure factor for a polymer melt composed of linear molecules of the same chemical identity but of two different chain lengths. The fluid is treated within a dynamical mean-field approach, in which each molecule is represented as a freely jointed chain moving among stochastic obstacles. The obstacles are of two types, each representing the obstruction of local conformational changes by one of the species present. The obstacle dynamics are determined self-consistently by equating the relaxation rate of an obstacle of a given type to the smallest conformational relaxation rate of the species that it represents. Calculation of the dynamic structure factor is mapped onto the solution of a random walk with dynamical disorder, in which a walker moves on a one-dimensional lattice with hopping rates that randomly fluctuate among three states. The relevant random walk problem is solved within the effective medium approximation, and the results are employed to examine the dependence of the dynamic structure factor on time, wave vector, chain lengths, and fluid composition.

Matrix method for random walks on lattices

By defining a structure factor matrix we propose a generating function matrix method for studying random walks on a class of lattices, such as superlattices or crystals with finite k atoms per cell. As an illustration of this method, the problem of random walks on a one-dimensional dimerization chain is solved exactly.