Asymmetric Random Walk Research Articles

The ruin problem and cover times of asymmetric random walks and Brownian motions

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

The ruin problem and cover times of asymmetric random walks and Brownian motions

A simple asymmetric random walk on the integers is stopped when its range is of a given length. When and where is it stopped? Analogous questions can be stated for a Brownian motion. Such problems are studied using results for the classical ruin problem, yielding results for the cover time and the range, both for asymmetric random walks and Brownian motion with drift.

How many probes are needed to compute the maximum of a random walk?

Odlyzko (1995) proves that, in the average, c n + o( n ) probes are necessary to compute the maximum of a simple symmetric random walk with n steps, in which c appears under the form of a triple integral. In this paper we prove that ( log n/| log p− log q|)+ o( log n) probes are necessary to compute the maximum of a simple asymmetric random walk with n steps. We also give c under closed form.

Anomalous diffusion resulting from strongly asymmetric random walks

We present a model of one-dimensional asymmetric random walks. Random walkers alternate between flights (steps of constant velocity) and sticking (pauses). The sticking time probability distribution function (PDF) decays as $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\nu}}$. Previous work considered the case of a flight PDF decaying as $P(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\mu}}$ [Weeks et al., Physica D 97, 291 (1996)]; leftward and rightward flights occurred with differing probabilities and velocities. In addition to these asymmetries, the present strongly asymmetric model uses distinct flight PDFs for leftward and rightward flights: ${P}_{L}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\mu}}$ and ${P}_{R}(t)\ensuremath{\sim}{t}^{\ensuremath{-}\ensuremath{\eta}}$, with $\ensuremath{\mu}\ensuremath{\ne}\ensuremath{\eta}$. We calculate the dependence of the variance exponent $\ensuremath{\gamma}({\ensuremath{\sigma}}^{2}\ensuremath{\sim}{t}^{\ensuremath{\gamma}})$ on the PDF exponents $\ensuremath{\nu},\ensuremath{\mu}$, and $\ensuremath{\eta}$. We find that $\ensuremath{\gamma}$ is determined by the two smaller of the three PDF exponents, and in some cases by only the smallest. A PDF with decay exponent less than 3 has a divergent second moment, and thus is a L\'evy distribution. When the smallest decay exponent is between 3/2 and 3, the motion is superdiffusive $(1<\ensuremath{\gamma}<2)$. When the smallest exponent is between 1 and 3/2, the motion can be subdiffusive $(\ensuremath{\gamma}<1)$; this is in contrast with the case with \ensuremath{\mu}=\ensuremath{\eta}.

Range reliability in random walks

This paper introduces a definition of reliability based on a process range. Thus, process failure is defined when the range of a process first reaches a given and unacceptable level. The Mean Time To Failure (MTTF) which is denned as the mean of the first time for a range to attain a given amplitude is then calculated for an asymmetric random walk process. The probability distribution of the range is then given and the process reliability over long periods of system operations are then calculated. Applications such as the control of wings movements, stock price and exchange rates volatility (defined in terms of reliability) are also used to motivate the usefulness of range processes in reliability studies. Finally, we point out that there is necessarily a relationship between the range reliability and the propensity of a series to become chaotic.

Anomalous diffusion in asymmetric random walks with a quasi-geostrophic flow example

We present a model of one-dimensional symmetric and asymmetric random walks. The model is applied to an experiment studying fluid transport in a rapidly rotating annulus. In the model, random walkers alternate between flights (steps of constant velocity) and sticking (pauses between flights). Flight time and sticking time probability distribution functions (PDFs) have power law decays: P( t) ∼ t − μ and t − ν for flights and sticking, respectively. We calculate the dependence of the variance exponent γ ( σ 2 ∼ t γ ) on the PDF exponents μ and ν. For a broad distribution of flight times ( μ < 3), the motion is superdiffusive (1 < γ < 2), and the PDF has a divergent second moment, i.e., it is a Lévy distribution. For a broad distribution of sticking times ( ν < 3), either superdiffusion or subdiffusion ( γ < 1) can occur, with qualitative differences between symmetric and asymmetric walks. For narrow PDFs ( μ > 3, ν > 3), normal diffusion ( γ = 1) is recovered. Predictions of the model are related to experimental observations of transport in a rotating annulus. The Eulerian velocity field is chaotic, yet it is still possible to distinguish between well-defined sticking events (particles trapped in vortices) and flights (particles making long excursions in a jet). The distribution of flight lengths is well described by a power law with a divergent second moment (Lévy distribution). The observed transport is strongly asymmetric and is well described by the proposed model.

Random walk on the Bethe lattice and hyperbolic Brownian motion

We give the exact solution to the problem of a random walk on the Bethe lattice through a mapping on an asymmetric random walk on the half-line. We also study the continuous limit of this model, and discuss in detail the relation between the random walk on the Bethe lattice and Brownian motion on a space of constant negative curvature.

Bistability for asymmetric discrete random walks

We show that asymmetric time-continuous discrete random walks can display bistability for equal values of Jauslin's shifting parameters. The bistability becomes more pronounced at increased asymmetry parameter

A matrix equation approach to solving recurrence relations in two-dimensional random walks

The purpose of this paper is to provide explicit formulas for a variety of probabilistic quantities associated with an asymmetric random walk on a finite rectangular lattice with absorbing barriers. Quantities of interest include probabilities that a walker will exit the lattice onto some particular set of boundary states, the expected duration of the walk, and the expected number of visits to one state given a start in another. These quantities are shown to satisfy two-dimensional recurrence relations that are very similar in structure. In each case, the recurrence relations may be represented by matrix equations of the form X = AX + XB + C, where A and B are tridiagonal Toeplitz matrices. The spectral properties of A and B are investigated and used to provide solutions to this matrix equation. The solutions to the matrix equations then lead to solutions for the recurrence relations in very general cases.

On multiple phase transitions for branching Markov chains

We consider branching Markov chains on a countable set. We give a necessary and sufficient condition in terms of the transition kernel of the underlying Markov chain to have two phase transitions. We compute the critical values. We apply this result to prove that asymmetric branching random walks onZ have two phase transitions.

Reaction of Paramecium to Temperature Gradients and the Telegraph Equation

The telegraph equation with a drift term to determine an asymmetric random walk is applied to the behavior of paramecia to accumulate at their optimal temperature in the presence of temperature gradient. From this equation, the following two points are clarified; 1, the stationary solution of the equation explains that the observed stationary distribution of the cells decreases exponentially from their optimal position, and 2, on a kind of paramecium that begins to accumulate at the optimal temperature after the treatment with riboflavin, the calculated transition time from a random distribution to the stationary distribution agrees with the observed transition time, 3 minutes.

Average versus typical mean first-passage time in an asymmetric random walk in a random medium.

The asymptotic behavior of the first-passage time on a one-dimensional random chain with 〈ln(p)〉&gt;〈ln(q)〉 but 〈q/p〉&gt;1 is investigated (p and q are the transition probabilities of moving to the right and left, respectively). It is shown that the average and typical first-passage times differ. Asymptotically the first-passage time to a specified site at a distance L to the right of the starting location varies as ${\mathit{L}}^{\mathit{x}}$ (x&gt;1), with a probability of 1 even though the average first-passage time goes as exp(CL).

Asymmetric random walk on a random Thue-Morse lattice

We study the behavior of an asymmetric random walk in a one-dimensional environment whose nonuniformity is in between that of quasi-periodic and random. We construct the environment from arithmetic subsequences of the Thue-Morse sequence. The construction induces in a natural way a measure μ on the space of environments which is invariant and ergodic with respect to translations but is not mixing and has zero entropy. The behavior of the random walk is rather similar to that found by Sinai for the Bernoulli case, when μ is a product measure for which the entropy has its maximum value; i.e. the particle motion is subdiffusive, the displacement growing in time as (log)t)1β, β = log 3/log4. The nature of the dramatic Sinai-Golosov “localization” is however quite different, exhibiting an interesting fractal structure whose nature depends upon the time scale of observation.

A Kinetic Model for the Energy Transfer in Phycobilisomes

A kinetic model for the energy transfer in phycobilisome (PBS) rods of Synechococcus 6301 is presented, based on a set of experimental parameters from picosecond studies. It is shown that the enormous complexity of the kinetic system formed by 400-500 chromophores can be greatly simplified by using symmetry arguments. According to the model the transfer along the phycocyanin rods has to be taken into account in both directions, i.e., back and forth along the rods. The corresponding forward rate constants for single step energy transfer between trimeric disks are predicted to be 100-300 ns(-1). The model that best fits the experimental data is an asymmetric random walk along the rods with overall exciton kinetics that is essentially trap-limited. The transfer process from the sensitizing to the fluorescing C-PC phycocyanin chromophores (tau approximately 10 ps) is localized in the hexamers. The transfer from the innermost phycocyanin trimer to the core is calculated to be in the range 36-44 ns(-1). These parameters lead to calculated overall rod-core transfer times of 102 and 124 ps for rods containing three and four hexamers, respectively. The model calculations confirm the previously suggested hypothesis that the energy transfer from the rods to the core is essentially described by one dominant exponential function. Extension of the model to heterogeneous PBS rods, i.e., PBS containing also phycoerythrin, is straightforward.

Asymmetric random walk model for laser induced drift

A theory is presented for laser induced drift based on an asymmetric random walk. The random walk formulation generates an intensity dependent drift velocity, which is valid far from any boundaries.

Phase transition in one-dimensional random walk with partially reflecting boundaries

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

Asymptotic survival probability of random walks on a semi-infinite line

Symmetric and asymmetric random walks on a segment (−∞,T>0) of the real line are considered. There is a non-zero probability for the random walk to get absorbed at a site it visits. We derive for such random walks, expressions for survival probabilities in the asymptotic limit ofT→∞. An application of this asymptotic formulation to the problem of radiation transport through thick shields is presented.

Phase transition in one-dimensional random walk with partially reflecting boundaries

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

Biological organization, Jarkievich measures and asymmetric random walks

In this paper we analyze the organization imposed by the energy input during the migration of enzymes on DNA. We attempt to measure that organization by means of a concept proposed by A.A. Jarkievich in 1961. We found relationships among a Jarkievich measure, the energy dissipation, and the fluctuations in the kinematic velocity of the enzyme on the DNA.

A Statistical Study of Fish Movement in an Aquarium

In order to observe the motion and behavior of a fish statistically, an optical apparatus connected to a microcomputer was designed. Using this apparatus, the position of a goldfish at an arbitrary time was measured in an aquarium and the one-dimensional distributions of the position and the speed were obtained. The experimental results show that the position distribution decreases exponentially with respect to increasing distance from the wall, while the speed distribution is independent of the position in the aquarium. Assuming one-dimensional asymmetric random walk and complete reflection at the wall, the exponential distribution of the fish's position is predicted theoretically.