Type Of Random Walk Research Articles

Dynamic Efficiency in the East European Emerging Markets

The paper re-investigates the efficiency of the East European emerging markets of the Czech Republic, Hungary, Poland and Russia analyzed by Rockinger and Urga (2000, 2001) based on the data from September, 1995 through December, 2004. We propose a first-order autoregressive (AR (1)) type time varying parameter model with a non-stochastic linear time trend including the random walk (RW) type model as a special case. The observed data rejects the RW type model for the AR (1) type one. The markets exhibit dynamic efficiency for all the four countries in the sense that the linear time trend approaches to zero over time. The empirical result for the Russian markets differs from that of Rockinger and Urga (2000).

Brownian motion of liquid lead inclusions along dislocations in aluminum

Nanosized solid Pb inclusions in Al have fcc structure and perfect cuboctahedral shape. When the inclusions melt they attain a rounded equilibrium shape and due to the low miscibility between the two elements, the liquid inclusions are retained as individual nanosized droplets located inside the solid Al matrix. As the temperature is increased the smaller liquid Pb inclusions are observed to move in a three-dimensional random walk over distances that can be orders of magnitude larger than their size. Inclusions attached to dislocations exhibit a similar type of random walk confined to long-distance one-dimensional movement along the dislocation lines and short-range two-dimensional vibrations perpendicular to the dislocation lines. The diffusion coefficients of the moving inclusions can be obtained from an Einstein-Smoluchowski analysis of the inclusion traces providing input for information on the mechanisms responsible for the movements.

Bad configurations for random walk in random scenery and related subshifts

In this paper we consider an arbitrary irreducible random walk on Z d , d ⩾ 1 , with i.i.d. increments, together with an arbitrary i.i.d. random scenery. Walk and scenery are assumed to be independent. Random walk in random scenery (RWRS) is the random process where time is indexed by Z , and at each unit of time both the step taken by the walk and the scenery value at the site that is visited are registered. Bad configurations for RWRS are the discontinuity points of the conditional probability distribution for the configuration at the origin of time given the configuration at all other times. We show that the set of bad configurations is non-empty. We give a complete description of this set and compute its probability under the random scenery measure. Depending on the type of random walk, this probability may be zero or positive. For simple symmetric random walk we get three different types of behavior depending on whether d = 1 , 2 , d = 3 , 4 or d ⩾ 5 . Our classification is actually valid for a class of subshifts having a certain determinative property, which we call specifiable, of which RWRS is an example. We also consider bad configurations w.r.t. a finite time interval (replacing the origin) and obtain an almost complete generalization of our results. Remarkably, this extension turns out to be somewhat delicate.

Random walks and chemical graph theory.

Simple random walks probabilistically grown step by step on a graph are distinguished from walk enumerations and associated equipoise random walks. Substructure characteristics and graph invariants correspondingly defined for the two types of random walks are then also distinct, though there often are analogous relations. It is noted that the connectivity index as well as some resistance-distance-related invariants make natural appearances among the invariants defined from the simple random walks.

The Filling Potential Method: A Method for Estimating the Free Energy Surface for Protein−Ligand Docking

We developed a molecular simulation method suitable for estimation of binding free energy, called the filling potential method, based on the concept of the Taboo search, which is a type of self-avoiding random walk consisting of a cycle of local-minimum searches and transition state searches. The filling potential method is an umbrella potential sampling method, and enables the ligand molecule to drift from its local minima automatically. In the case of the filling potential method, the umbrella potential is a combination of Gaussian-type repulsive potentials, which are located on the trajectory of the ligand. Without setting the reaction coordinates a priori, this method searches for and determines the suitable reaction coordinates by successive generation of umbrella potentials based on its trajectory analysis. The weighted histogram analysis for these trajectories gives the binding free energy of the ligand to the receptor protein. It was applied to the complex of thermolysin and its inhibitors in expl...

Area of Brownian Motion with Generatingfunctionology

This paper gives a survey of the limit distributions of the areas of different types of random walks, namely Dyck paths, bilateral Dyck paths, meanders, and Bernoulli random walks, using the technology of generating functions only.

Limit theorems and absorption problems for quantum random walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS...

Computing stationary probability distributions and large deviation rates for constrained random walks.

Our model is a constrained homogeneous random walk in Z + d . The convergence to stationarity for such a random walk can often be checked by constructing a Lyapunov function. The same Lyapunov function can also be used for computing approximately the stationary distribution of this random walk, using methods developed in [11]. In this paper we show that computing exactly the stationary probability for this type of random walks is an undecidable problem: no algorithm can exist to achieve this task. We then prove that computing large deviation rates for this model is also an undecidable problem. We extend these results to a certain type of queueing systems. The implication of these results is that no useful formulas for computing stationary probabilities and large deviations rates can exist in these systems.

Limit theorems and absorption problems for quantum radom walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by $2 \times 2$ unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

Fractal dimensions and random walks on random trees

We determine the fractal dimension d f of infinite spherically symmetric random trees (all vertices at distance n from the root have the same degree d n where { d n } are independent random variables). If d n takes the value 3 or 2 with probabilities q n and 1− q n , then d f =( log 2) lim n nq n+1 a.s. We show how d f is closely related to the type of the simple random walks (SRW) on trees. We prove that the SRW is a.s. transient if d f>2 a.s. and a.s. recurrent if d f<2 a.s. and if d f=2 a.s. we obtain a.s. transience or recurrence. We also consider another type of random trees which are corresponding to branching processes in varying environments. In particular, we consider a tree such that the degrees of the vertices at distance n from the root are independent identically distributed (iid) random variables following the distribution of a random variable that takes the value 3 or 2 with probabilities q n and 1− q n respectively. These iid random variables are also independent of the degrees of the vertices of the other generations. We prove that the SRW is a.s. recurrent if and only if d f⩽2 a.s. We also prove for such trees that d f = lim n nq n+1 a.s.

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.

Heterogeneous tube model for the study of small intestinal transit flow.

A Monte-Carlo computer simulation technique was employed to study the details of the small intestinal transit flow in the gastrointestinal (GI) tract. A heterogeneous tube model was constructed using a numerical computer simulation technique. The model was built from first principles and included several heterogeneous characteristics of the GI tract structure. We used a random, dendritic-type internal structure representing the villi of the GI tract. The small intestinal transit flow was simulated using two diffusion models, namely, the blind ant and the myopic ant models, which are different models to account the elapse of time, and which are both based on statistical properties of random walks. For each one of the models we utilize two types of biased random walk, placing different emphasis in the motion towards the output of the tube. We monitored the flow of the drug in terms of Monte-Carlo time steps (MCS) through the tube walls and dendritic villi present. The frequency of the transit times was dependent on the structure of the dendritic villi and on the type of biased random walk. The small intestinal flow profile of literature data for a large number of drugs was well characterized by the heterogeneous model using, as parameters, a certain number of villi per unit length of the tube and specific characteristics for both types of the biased random walk. A correspondence between the MCS and real time units was achieved. The transit process of the oral dosage forms in the GI tract can be reproduced with the heterogeneous model developed. This model can be used to study GI absorption phenomena.

Tauberian Theorems via Statistical Convergence

The concept of statistical convergence, which is related to the usual concept of convergence in probability, provides a regular summability method for abstract metric spaces. By using probabilistic tools, we provide some Tauberian theorems which have best possible order Tauberian conditions. Furthermore, these methods can be used to unify and improve the classical pointwise Tauberian theorems of summability theory for the random walk type methods as proved by Bingham, and Hausdorff methods as proved by Lorentz.

An Algorithm and a C-program Source for Automated Drainage Network Extraction

We developed a flooding-type algorithm for drainage network extraction and show a full program source in the language C. The program generates automatically Drainage Direction Matrices (DDMs) from Digital Elevation Models (DEMs) with sink holes. By using a random technique of random walk type, the drainage networks are processed adequately in flat plains or alluvial fans.

Shapes and sizes of arbitrary random walks at O(1/ d3) II. Asphericity and prolateness parameters

For arbitrary random walks in any d-dimensional space, expansions in powers of 1/ d of asphericity and prolateness parameters and moments of the inverse size ratio have been developed, which, at O(1/ d 3), yield very good approximations to exact values of the parameters for chains, rings, dumbbells and 3- and 5-arm regular stars. The 1/ d-expansions have also been used to obtain an estimate of these shape asymmetry parameters for 3D Edwards chains, rings, dumbbells and 3-arm stars and to give a mathematical proof that infinitely large random nets such as Bethe lattice or starburst and Mckay's net exhibit spherical symmetry. For arbitrary random walks at d = ∞, it is proved that these parameters coincide with their corresponding factors, while for an end-looped self-avoiding walk, it is found that its shape asymmetry is even larger than that of an open SAW. An 1/ f-expansion of the parameters for f-arm regular stars has also been obtained, and a comparison of the dimensionality dependence of the parameters with that of the corresponding factors has been made for the four types of random walks.

Random walk in the Cu/graphite mixtures.

Transport properties of Cu/graphite composites scanned surfaces were studied by the random walk. The type of random walk used considers the nonzero conductivity of graphite. The obtained results are consistent with the existing scaling theory. The diffusivity composition dependence behavior in two dimensions is in good agreement with the measured electrical conductivity composition dependence behavior of Cu/graphite composites in three dimensions. \textcopyright{} 1996 The American Physical Society.

Shortest expected delay routing for Erlang servers

The queueing problem with Poisson arrivals and two identical parallel Erlang servers is analyzed for the case of shortest expected delay routing. This problem may be represented as a random walk on the integer grid in the first quadrant of the plane. An important aspect of the random walk is that it is possible to make large jumps in the direction of the boundaries. This feature gives rise to complicated boundary behavior. Generating function approaches to analyze this type of random walk seem to be extremely complicated and have not been successful yet. The approach presented in this paper directly solves the equilibrium equations. It is shown that the equilibrium distribution of the random walk can be written as an infinite linear combination of products. This linear combination is constructed in a compensation procedure. The starting solutions for this procedure are found by solving the shortest expected delay problem with instantaneous jockeying. The results can be used for an efficient computation of performance criteria, such as the waiting time distribution and the moments of the waiting time and the queue lengths.

The rate of convergence of a homogeneous Markov chain arising from two-queue networks

This paper is concerned with the type of random walks in two dimensions considered by Malyshev [1], which describe a wide class of two-queue systems. Sufficient conditions are established for the random walks to have various rate of convergence results including ergodicity, geometric ergodicity and sub-geometric rates for a variety of rate functions.

On three conjectures by K. E. Shuler

Some fifteen years ago, Shuler formulated three conjectures relating to the large-time asymptotic properties of a nearest-neighbor random walk on ℤ2 that is allowed to make horizontal steps everywhere but vertical steps only on a random fraction of the columns. We give a proof of his conjectures for the situation where the column distribution is stationary and satisfies a certain mixing codition. We also prove a strong form of scaling to anisotropic Brownian motion as well as a local limit theorem. The main ingredient of the proofs is a large-deviation estimate for the number of visits to a random set made by a simple random walk on ℤ. We briefly discuss extensions to higher dimension and to other types of random walk.

Shape asymmetry of random walks and nonreversal random walks

Random walks (RWs) and nonreversal random walks (NRRWs) embedded in various lattices and freely jointed (off-lattice) chains—consisting of up to N≊1000 segments—have been produced and analyzed with respect to their instantaneous shape. While the results of different RWs (as expected) coincide for all chain-lengths examined, the short-chain behavior of NRRWs is strongly dependent on the lattice type. In the limit of infinitely long chains, however, quantities characteristic of the shape converge to common values for all types of RWs and NRRWs examined.