Dimensional Random Walk Research Articles

Monotonicity for excited random walk in high dimensions

We prove that the drift θ(d, β) for excited random walk in dimension d is monotone in the excitement parameter \({\beta \in [0,1]}\) , when d is sufficiently large. We give an explicit criterion for monotonicity involving random walk Green’s functions, and use rigorous numerical upper bounds provided by Hara (Private communication, 2007) to verify the criterion for d ≥ 9.

Quenched limits for transient, zero speed one-dimensional random walk in random environment

We consider a nearest-neighbor, one dimensional random walk {Xn}n≥0 in a random i.i.d. environment, in the regime where the walk is transient but with zero speed, so that Xn is of order ns for some s<1. Under the quenched law (i.e., conditioned on the environment), we show that no limit laws are possible: There exist sequences {nk} and {xk} depending on the environment only, such that Xnk−xk=o(log nk)2 (a localized regime). On the other hand, there exist sequences {tm} and {sm} depending on the environment only, such that logsm/log tm→s<1 and Pω(Xtm/sm≤x)→1/2 for all x>0 and →0 for x≤0 (a spread out regime).

Rain attenuation time series synthesis with simulated rain cell movement

This paper proposes a stochastic modeling method to simulate the rain attenuation process in order to generate time series for high frequency terrestrial radio links. It is also the subject of this research to express the local attenuation differences on converging radio links belonging to the same node. The simulation scene (some tenth square kilometers) is a cellular mobile radio system which consists of a backbone node and some millimeter band radio links forming a star network around the node. The model parameters are extracted from the long-term measurement of different local meteorological parameters of an active radio network. The essentials of the model is the simulation of rain cell translation across the scene by a two dimensional non-symmetrical random walk (discrete time and discrete state homogenous Markov model). A second Markov model is applied to embed in the model the rain cell translation speed. The structure of the rain cells is approximated with an elliptical model to express the point rai nfall rate in order to calculate the path attenuation at discrete time steps during the simulation. The local precipitation field allows the evaluation of the rainfall impact on the considered radio links in terms of single-link attenuation distributions. The model is also applicable to simulate the annual attenuation statistics which will be compared with the real measurement statistics for evaluation purposes.

Superdiffusion in a class of networks with marginal long-range connections

A class of cubic networks composed of a regular one-dimensional lattice and a set of long-range links is introduced. Networks parametrized by a positive integer k are constructed by starting from a one-dimensional lattice and iteratively connecting each site of degree 2 with a kth neighboring site of degree 2. Specifying the way pairs of sites to be connected are selected, various random and regular networks are defined, all of which have a power-law edge-length distribution of the form P_{>}(l) approximately l;{-s} with the marginal exponent s=1 . In all these networks, lengths of shortest paths grow as a power of the distance and random walk is superdiffusive. Applying a renormalization group method, the corresponding shortest-path dimensions and random-walk dimensions are calculated exactly for k=1 networks and for k=2 regular networks; in other cases, they are estimated by numerical methods. Although, s=1 holds for all representatives of this class, the above quantities are found to depend on the details of the structure of networks controlled by k and other parameters.

Probably not: future prediction using probability and statistical inference

Introduction. 1. An Introduction to Probability. Predicting The Future. Rule Making. Random Events and Probability. The Lottery {Very Improbable Events and Large Data Sets}. Coin Flipping {Fair Games, Looking Backwards For Insight}. The Coin Flip Strategy That Can't Lose. The Prize Behind The Door {Looking Backwards For Insight, Again}. The Checker Board {Dealing With Only Part Of The Data Set}. 2. Probability Distribution Functions And Some Basics. The Probability Distribution Function. Averages And Weighted Averages. Expected Values. The Basic Coin Flip Game. The Standard Deviation. The Cumulative Distribution Function. The Confidence Interval. Final Points. 3. Building a Bell. 4. Random Walks. The One Dimensional Random Walk. What Probability Really Means. Diffusion. 5. Life Insurance and Social Security. Insurance as Gambling. Life Tables. Birth Rates and Population Stability. Life Tables, Again. Premiums. Social Security - Sooner Or Later?. 6. Binomial Probabilities. The Binomial Probability Formula. Permutations And Combinations. Large Number Approximations. The Poisson Distribution. Disease Clusters. Clusters. 7. Pseudorandom Numbers and Monte Carlo Simulations. Pseudorandom Numbers. The Middle Square PSNG. The Linear Congruential PSNG. A Normal Distribution Generator. An Arbitrary Distribution Generator. Monte Carlo Simulations. A League Of Our Own. 8. Some Gambling Games In Detail. The Basic Coin Flip Game. The Gantt Chart. The Ultimate Winning Strategy. The Game Show. Parimutuel Betting. 9. Traffic Lights And Traffic. Outsmarting A Traffic Light?. Many Lights And Many Cars. Simulating Traffic Flow - The Simulation. Simulation Results. 10. Combined And Conditional Probabilities. Functional Notation. Conditional Probability. Medical Test Results. The Shared Birthday Problem. 11. Scheduling And Waiting. Scheduling Appointments In The Doctor's Office. Lunch With A Friend. Waiting For A Bus. 12. Stock Market Portfolios. 13. Benford, Parrondo and Simpson. Benford's Law. Parrondo's Paradox. Simpson's Paradox. 14. Networks, Infectious Disease Propagation and Chain Letters. Degrees Of Separation. Propagation Along Networks. Some Other Uses Of Networks. Neighborhood Chains. 15. Bird Counting. A Walk In The Woods. A Model Of Bird Flying Habits. Spotting A Bird. Putting It All Together. 16. Statistical Mechanics And Heat. Statistical Mechanics. Thermodynamics. 17. Introduction To Statistical Analysis. Sampling. Sample Distributions and Standard Deviations. Estimating Population Average From A Sample. The Student-T Distribution. Polling Statistics. Did A Sample Come From A Given Population?. 18. Chaos and Quanta. Chaos. Probability In Quantum Mechanics.

A functional CLT for the occupation time of a state-dependent branching random walk

We show that the centred occupation time process of the origin of a system of critical binary branching random walks in dimension d≥3, started off either from a Poisson field or in equilibrium, when suitably normalized, converges to a Brownian motion in d≥4. In d=3, the limit process is a fractional Brownian motion with Hurst parameter 3/4 when starting in equilibrium, and a related Gaussian process when starting from a Poisson field. For (dependent) branching random walks with state dependent branching rate we obtain convergence in f.d.d. to the same limit process, and for d=3 also a functional limit theorem.

On the nature of the heterogeneous dynamics of ions in ionic conducting glasses

Molecular dynamics simulations were performed to study the complex and heterogeneous dynamics of ions in ionically conducting glasses. The dynamics of Li ions in lithium silicate glasses have been characterized by van-Hove functions, fractal dimension of random walks, multifractal analysis of density profile, and principal component analyses of time series. Hierarchy dynamics with local and successive jumps, which are followed by cooperative jumps involving more ions, are observed in the self-part of the van-Hove functions and in trajectories where they form a multifractal density profile. Fractal dimensions of the random walks can be used to characterize the motions of ions in both the short and long length-scale regions. Difference between them becomes clearer in the glassy state, and the corresponding power law dependences of the mean squared displacements can be well explained by these exponents. Phase-space plot using the de-noised data obtained by principal component analysis reveals that the trajectory extending from the short time to the long time dynamics in the plot is continuous and has long term memory. The motion examined by the principal component analysis of the time series shows the deterministic character of the single particle ion dynamics. Each ion moves among domains of collective motion. Recurrence plot and mutual information function shows long correlation of the dynamics. Therefore, the cause of the dynamic heterogeneity is rather deterministic in nature.

Large deviations estimates for self-intersection local times for simple random walk in $${\mathbb{Z}}^3$$

We obtain large deviations estimates for the self-intersection local times for a simple random walk in dimension 3. Also, we show that the main contribution to making the self-intersection large, in a time period of length n, comes from sites visited less than some power of log(n). This is opposite to the situation in dimensions larger or equal to 5. Finally, we present an application of our estimates to moderate deviations for random walk in random sceneries.

A functional limit theorem for a 2d-random walk with dependent marginals

We prove a non-standard functional limit theorem for a two dimensional simple random walk on some randomly oriented lattices. This random walk, already known to be transient, has different horizontal and vertical fluctuations leading to different normalizations in the functional limit theorem, with a non-Gaussian horizontal behavior. We also prove that the horizontal and vertical components are not asymptotically independent.

A multifractal decomposition according to rate of returns

AbstractFor one‐dimensional simple symmetric random walk, the Hausdorff and packing dimensions of sets of sample paths with prescribed rate of returns to the origin are determined. This gives a multifractal decomposition of the underlying sample space. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Central Limit Theorem For The Excited Random Walk In Dimension $d\geq 2$

We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d\geq 2$.

SIGEST

How often is it the case that not only the applications but also the methods of analysis of an important mathematical property range all the way from the deeply mathematical to the realm of the social sciences (and beyond)? Or, for that matter, that one can learn about the property under consideration in some detail on Wikipedia? The {small world phenomenon} is an instance of all of these characteristics.For the mathematical community, the small world phenomenon came alive with the 1998 paper of Watts and Strogatz that looked at the phenomenon from the perspective of a lattice network model and showed that the addition of a relatively small number of randomly chosen long—range connections to an otherwise locally connected lattice leads to a very good model of the small world phenomenon. Their work and others showed further that this type of model describes the actual connectivity and behavior of applications in areas as diverse as the brain of a small organism, the electrical power grid, and connectivity on the World Wide Web.This issue's SIGEST paper, ’’A Matrix Perturbation View of the Small World Phenomenon’’ by D. J. Higham that originally was published in the { SIAM Journal on Matrix Analysis and Applications} in 2003, makes an inspired and important contribution to the mathematical understanding of the small world phenomenon by looking at it from a new perspective. As Higham discusses in the new preamble to the paper, he was motivated to formulate a model that could display the small world phenomenon and be amenable to rigorous mathematical analysis. He succeeded by considering a Markov chain that also can be thought of as a one—dimensional periodic random walk, where the standard connections to the two nearest neighbors are augmented by low probability uniform jumps to any state, also referred to as shortcuts. Higham uses matrix perturbation theory methods to study the correspondence between the probabilities of these uniform jumps and the mean hitting time (the average number of steps to get from some random state to a given fixed state) of the random walk. In particular, the mean hitting times of a Markov chain can be computed by solving a certain system of linear equations, and by use of matrix perturbation theory, Higham analyzes how these times vary with the probability of the uniform jumps. The paper shows that this model is very effective in explaining the small world phenomenon mathematically, and that in addition, its mathematical behavior corroborates behavior that was observed experimentally by Watts and Strogatz. Higham's model also has a close connection to techniques used in Internet search engines.We hope that many SIAM readers will enjoy this interesting, accessible, and very nicely written paper that adds great insight into an important property of the natural and social worlds. And those of you with Hollywood connections will have something novel to bring up the next time the conversation turns to the ’’Six Degrees of Kevin Bacon.’’

Anomalous diffusion in two‐dimensional Euclidean and prefractal geometrical models of heterogeneous porous media

Disordered systems are known to induce anomalous diffusion. This phenomenon may be important for environmental applications such as contaminant transport and nutrient availability. However, few studies have investigated anomalous diffusion in this context. In particular, the relationship between pore space geometry and anomalous diffusion is not well understood. We report on numerical simulations of solute diffusion within the water‐filled pore spaces of two‐dimensional geometrical models of heterogeneous porous media. Euclidean and mass, pore, and pore‐solid prefractal lattices were used to generate random pore networks with varying porosity (ϕ) and lacunarity (L). The objectives were to investigate the effects of ϕ and L on the solute random walk dimension (dw) and to identify which of these models best represents a natural porous medium. Solute diffusion was simulated using a stochastic cellular automaton based on the “myopic ant” algorithm. Estimates of dw ≫ 2 occurred with increasing frequency as ϕ → 0, indicating scale dependency in the standard diffusion coefficient at low porosities. The relationship between dw and ϕ for the mass and pore‐solid prefractal lattices was the closest to that for natural 2‐D systems (i.e., soil thin sections). The presence of large, interconnected pore spaces (L → 1) at low porosities reduced the intensity of anomalous diffusion (dw → 2). A power law relationship based on the product of dw and L explained &gt;96% of the total variation in ϕ regardless of the type of lattice considered. The potential predictive capability of this approach for natural porous media deserves further investigation.

Simple transient random walks in one-dimensional random environment: the central limit theorem

We consider a simple random walk (dimension one, nearest neighbour jumps) in a quenched random environment. The goal of this work is to provide sufficient conditions, stated in terms of properties of the environment, under which the central limit theorem (CLT) holds for the position of the walk. Verifying these conditions leads to a complete solution of the problem in the case of independent identically distributed environments as well as in the case of uniformly ergodic (and thus also weakly mixing) environments.

Observation of anomalous diffusion in excised tissue by characterizing the diffusion-time dependence of the MR signal

This report introduces a novel method to characterize the diffusion-time dependence of the diffusion-weighted magnetic resonance (MR) signal in biological tissues. The approach utilizes the theory of diffusion in disordered media where two parameters, the random walk dimension and the spectral dimension, describe the evolution of the average propagators obtained from q-space MR experiments. These parameters were estimated, using several schemes, on diffusion MR spectroscopy data obtained from human red blood cell ghosts and nervous tissue autopsy samples. The experiments demonstrated that water diffusion in human tissue is anomalous, where the mean-square displacements vary slower than linearly with diffusion time. These observations are consistent with a fractal microstructure for human tissues. Differences observed between healthy human nervous tissue and glioblastoma samples suggest that the proposed methodology may provide a novel, clinically useful form of diffusion MR contrast.

A two-parameter random walk with approximate exponential probability distribution

We study a non-Markovian random walk in dimension 1. It depends on two parameters eps_r and eps_l, the probabilities to go straight on when walking to the right, respectively to the left. The position x of the walk after n steps and the number of reversals of direction k are used to estimate eps_r and eps_l. We calculate the joint probability distribution p_n(x,k) in closed form and show that, approximately, it belongs to the exponential family.

Monte carlo simulation of diffusion-limited drug release from finite fractal matrices

How fast can drug molecules escape from a controlled matrix-type release system? This important question is of both scientific and practical importance, as increasing emphasis is placed on design considerations that can be addressed only if the physical chemistry of drug release is better understood. In this work, this problem is studied via Monte Carlo computer simulations. The drug release is simulated as a diffusion-controlled process. Six types of Menger sponges (all having the same fractal dimension, df = 2.727, but with different values of random walk dimension, dw ∈ [2.028, 2.998]) are employed as models of drug delivery devices with the aim of studying the consequences of matrix structural properties (characterized by df and dw) on drug release performance. The results obtained show that, in all cases, drug release from Menger sponges follows an anomalous behavior. Finally, the influence of the matrix structural properties on the drug release profile is quantified.

Breakdown of metastable step-flow growth on vicinal surfaces induced by nucleation

We consider the growth of a vicinal crystal surface in the presence of a step-edge barrier. For any value of the barrier strength, measured by the length ${\ensuremath{\ell}}_{\mathrm{ES}}$, nucleation of islands on terraces is always able to destroy asymptotically step-flow growth. The breakdown of the metastable step-flow occurs through the formation of a mound of critical width proportional to ${L}_{c}\ensuremath{\sim}1∕\sqrt{{\ensuremath{\ell}}_{\mathrm{ES}}}$, the length associated to the linear instability of a high-symmetry surface. The time required for the destabilization grows exponentially with ${L}_{c}$. Thermal detachment from steps or islands, or a steeper slope increase the instability time but do not modify the above picture, nor change ${L}_{c}$ significantly. Standard continuum theories cannot be used to evaluate the activation energy of the critical mound and the instability time. The dynamics of a mound can be described as a one dimensional random walk for its height $k$; attaining the critical height (i.e., the critical size) means that the probability to grow $(k\ensuremath{\rightarrow}k+1)$ becomes larger than the probability for the mound to shrink $(k\ensuremath{\rightarrow}k\ensuremath{-}1)$. Thermal detachment induces correlations in the random walk, otherwise absent.

Heavy points of a [formula omitted]-dimensional simple random walk

For a simple symmetric random walk in dimension d ⩾ 3 , a uniform strong law of large numbers is proved for the number of sites with given local time up to time n.

Limit theorems for U-statistics indexed by a one dimensional random walk

Let (Sn)n≥0 be a -random walk and be a sequence of independent and identically distributed -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on with values in . We study the weak convergence of the sequence , with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by Statistical applications are presented, in particular we prove a strong law of large numbers for U -statistics indexed by a one-dimensional random walk using a result of [1].