Stochastic flows,reaction — Diffusion processes,and morphogenesis

Recently, anexact procedure has been introduced [C. A. Walsh and J. J. Kozak,Phys. Rev. Lett. 47:1500 (1981)] for calculating the expected walk length 〈n〉 for a walker undergoing random displacements on a finite or infinite (periodic)d-dimensional lattice with traps (reactive sites). The method (which is based on a classification of the symmetry of the sites surrounding the central deep trap and a coding of the fate of the random walker as it encounters a site of given symmetry) is applied here to several problems in lattice statistics for each of whichexact results are presented. First, we assess the importance of lattice geometry in influencing the efficiency of reaction-diffusion processes in simple and multiple trap systems by reporting values of 〈n〉 for square (cubic) versus hexagonal lattices ind=2, 3. We then show how the method may be applied to variable-step (distance-dependent) walks for a single walker on a given lattice and also demonstrate the calculation of the expected walk length 〈n〉 for the case of multiple walkers. Finally, we make contact with recent discussions of “mixing” by showing that the degree of chaos associated with flows in certain lattice systems can be calibrated by monitoring the lattice walks induced by the Poincaré map of a certain parabolic function.     

Skyrmion burst and multiple quantum walk in thin ferromagnetic films

We propose a new type of quantum walk in thin ferromagnetic films. A giant Skyrmion collapses to a singular point in a thin ferromagnetic film, emitting spin waves, when external magnetic field is increased beyond the critical one. After the collapse the remnant is a quantum walker carrying spin S. We determine its time evolution and show the diffusion process is a continuous-time quantum walk. We also analyze an interference of two quantum walkers after two Skyrmion bursts. The system presents a new type of quantum walk for S>1/2, where a quantum walker breaks into 2S quantum walkers.     

Lattice random walks for sets of random walkers. First passage times

We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which sticks atD > 4.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE78-21460.     

Biased trapping issue on weighted hierarchical networks

In this paper, we present trapping issues of weight-dependent walks on weighted hierarchical networks which are based on the classic scale-free hierarchical networks. Assuming that edge’s weight is used as local information by a random walker, we introduce a biased walk. The biased walk is that a walker, at each step, chooses one of its neighbours with a probability proportional to the weight of the edge. We focus on a particular case with the immobile trap positioned at the hub node which has the largest degree in the weighted hierarchical networks. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping issue. Let parameter a (0 < a < 1) be the weight factor. We show that the efficiency of the trapping process depends on the parameter a; the smaller the value of a, the more efficient is the trapping process.     

Universal fluctuations in the support of the random walk

A random walk starts from the origin of ad-dimensional lattice. The occupation numbern(x,t) equals unity if aftert steps site x has been visited by the walk, and zero otherwise. We study translationally invariant sumsM(t) of observables defined locally on the field of occupation numbers. Examples are the numberS(t) of visited sites, the areaE(t) of the (appropriately defined) surface of the set of visited sites, and, in dimension d=3, the Euler index of this surface. Ind≤ 3, theaverages - M(t) all increase linearly witht ast ® ∞. We show that in d=3, to leading order in an asymptotic expansion int, thedeviations from average ΔM(t) = M(t) -M(t) are, up to a normalization, allidentical to a single “universal” random variable. This result resembles an earlier one in dimensiond=2; we show that this universality breaks down ford>3.     

Non-linear continuous time random walk models

A standard assumption of continuous time random walk (CTRW) processes is that there are no interactions between the random walkers, such that we obtain the celebrated linear fractional equation either for the probability density function of the walker at a certain position and time, or the mean number of walkers. The question arises how one can extend this equation to the non-linear case, where the random walkers interact. The aim of this work is to take into account this interaction under a mean-field approximation where the statistical properties of the random walker depend on the mean number of walkers. The implementation of these non-linear effects within the CTRW integral equations or fractional equations poses difficulties, leading to the alternative methodology we present in this work. We are concerned with non-linear effects which may either inhibit anomalous effects or induce them where they otherwise would not arise. Inhibition of these effects corresponds to a decrease in the waiting times of the random walkers, be this due to overcrowding, competition between walkers or an inherent carrying capacity of the system. Conversely, induced anomalous effects present longer waiting times and are consistent with symbiotic, collaborative or social walkers, or indirect pinpointing of favourable regions by their attractiveness.     

Random walks on sparsely periodic and random lattices: I. Random walk properties from lattice bond enumeration

We have developed a new technique for calculating certain asymptotic random walk properties on sparsely periodic and related random lattices in two and three dimensions. This technique is based on an ansatz which relates the number of lattice bonds in “irreducible lattice fragments” to the number of steps along these bonds. We show that certain random walk properties can be calculated very simply on the basis of this ansatz and that they depend only on the density of bonds and not on the arrangement of the bonds within the lattice. The random walk properties calculated here (mean square displacements, number of distinct sites visited, probability of return to the origin) are in complete agreement with results obtained earlier via generating function techniques. A subsequent paper contains generating function calculations which verify a number of new results presented here, such as mean occupation frequency of lattice sites, and a proof of our basic assumption on the relation between the number of lattice bonds and random walk steps.     

Two Quantum Coins Sharing a Walker

For classical random walks, changing or not changing coins makes a trivial influence on the random-walk behaviors. In this paper, we investigate the quantum walk where a walker’s movement is controlled by two initially independent coins alternately partially or fully after each step. We observe that there exist complicated inter-coin correlations in the quantum walk. Specifically, we study the correlations of two coins by tracing out the walker, and analyze classical, general, and quantum correlations between two coins in terms of classical mutual information, quantum mutual information, and measurement-induced disturbance. Our analysis shows different quantum features from that in classical random walks.

     

An exponential upper bound for the survival probability in a dynamic random trap model

We consider a symmetric translation-invariant random walk on thed-dimensional lattice ? ^d . The walker moves in an environment of moving traps. When the walker hits a trap, he is killed. The configuration of traps in the course of time is a reversible Markov process satisfying a level-2 large-deviation principle. Under some restrictions on the entropy function, we prove an exponential upper bound for the survival probability, i.e., $$\mathop {lim sup}\limits_{t \to \infty } \frac{1}{t}\log \mathbb{P}(T \geqslant t)< 0$$ whereT is the survival time of the walker. As an example, our results apply to a random walk in an environment of traps that perform a simple symmetric exclusion process.     

Propriétés d'intersection des marches aléatoires

We study intersection properties of multi-dimensional random walks. LetX andY be two independent random walks with values in ? ^d (d≦3), satisfying suitable moment assumptions, and letI _n denote the number of common points to the paths ofX andY up to timen. The sequence (I _n ), suitably normalized, is shown to converge in distribution towards the “intersection local time” of two independent Brownian motions. Results are applied to the proof of a central limit theorem for the range of a two-dimensional recurrent random walk, thus answering a question raised by N. C. Jain and W. E. Pruitt.     

Scaling relationships for anisotropic random walks

Aspects of transport in a highly multiple-scattering environment are investigated by examining random walkers moving in media having anisotropic angular scattering cross sections (turn-angle distributions). A general expression is obtained for the mean square displacement x² of a random walker executing ann-step walk in an infinite homogeneous material, and results are used to predict scaling relations for the probability() that a walker returns to the planar surface of a semi-infinite medium at a distance from the point of its insertion.     

Alzheimer random walk

Using the Monte Carlo simulation, we investigate a memory-impaired self-avoiding walk on a square lattice in which a random walker marks each of sites visited with a given probability p and makes a random walk avoiding the marked sites. Namely, p = 0 and p = 1 correspond to the simple random walk and the self-avoiding walk, respectively. When p> 0, there is a finite probability that the walker is trapped. We show that the trap time distribution can well be fitted by Stacy’s Weibull distribution \(b{\left( {\tfrac{a}{b}} \right)^{\tfrac{{a + 1}}{b}}}{\left[ {\Gamma \left( {\tfrac{{a + 1}}{b}} \right)} \right]^{ - 1}}{x^a}\exp \left( { - \tfrac{a}{b}{x^b}} \right)\) where a and b are fitting parameters depending on p. We also find that the mean trap time diverges at p = 0 as ~p ^{? α} with α = 1.89. In order to produce sufficient number of long walks, we exploit the pivot algorithm and obtain the mean square displacement and its Flory exponent ν(p) as functions of p. We find that the exponent determined for 1000 step walks interpolates both limits ν(0) for the simple random walk and ν(1) for the self-avoiding walk as [ ν(p) ? ν(0) ] / [ ν(1) ? ν(0) ] = p ^β with β = 0.388 when p ? 0.1 and β = 0.0822 when p ? 0.1.     

Single and multiple random walks on random lattices: Excitation trapping and annihilation simulations

Random walk simulations of exciton trapping and annihilation on binary and ternary lattices are presented. Single walker visitation efficiencies for ordered and random binary lattices are compared. Interacting multiple random walkers on binary and ternary random lattices are presented in terms of trapping and annihilation efficiencies that are related to experimental observables. A master equation approach, based on Monte Carlo cluster distributions, results in a nonclassical power relationship between the exciton annihilation rate and the exciton density.     

Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

Pressure effects on the absorption spectra of Na doped in solid Ar

1 Introduction Impelled by the research project for the High Energy Density Matter (HEDM) ofUSA, researchers have achieved fruitful results on the energetic species in cryogenicenvironments[1]. That the chemical performance of solid hydrogen fuel can be greatlyenhanced when being doped with a small amount of light metal is confirmed both theo-retically and experimentally[2]. For understanding the mechanism, similar systems haveattracted much interest. The system of rare gas solid doped w…     

Dynamical correlations for vicious random walk with a wall

A one-dimensional system of nonintersecting Brownian particles is constructed as the diffusion scaling limit of Fisher's vicious random walk model. N Brownian particles start from the origin at time t=0 and undergo mutually avoiding motion until a finite time t=T. Dynamical correlation functions among the walkers are exactly evaluated in the case with a wall at the origin. Taking an asymptotic limit N→∞, we observe discontinuous transitions in the dynamical correlations. It is further shown that the vicious walk model with a wall is equivalent to a parametric random matrix model describing the crossover between the Bogoliubov–deGennes universality classes.     

Adaptive random walks on the class of Web graphs

We study random walk with adaptive move strategies on a class of directed graphs with variable wiring diagram. The graphs are grown from the evolution rules compatible with the dynamics of the world-wide Web [B. Tadić, Physica A 293, 273 (2001)], and are characterized by a pair of power-law distributions of out- and in-degree for each value of the parameter β, which measures the degree of rewiring in the graph. The walker adapts its move strategy according to locally available information both on out-degree of the visited node and in-degree of target node. A standard random walk, on the other hand, uses the out-degree only. We compute the distribution of connected subgraphs visited by an ensemble of walkers, the average access time and survival probability of the walks. We discuss these properties of the walk dynamics relative to the changes in the global graph structure when the control parameter β is varied. For β≥ 3, corresponding to the world-wide Web, the access time of the walk to a given level of hierarchy on the graph is much shorter compared to the standard random walk on the same graph. By reducing the amount of rewiring towards rigidity limit β↦β_c≲ 0.1, corresponding to the range of naturally occurring biochemical networks, the survival probability of adaptive and standard random walk become increasingly similar. The adaptive random walk can be used as an efficient message-passing algorithm on this class of graphs for large degree of rewiring.     

Effect of particle size distribution and dynamics on the performance of two-dimensional packing

Extensive computer simulation is used to revisit and to generalize two classical problems: (i) the random car-parking dynamics of A. Rényi and (ii) the irreversible random sequential adsorption (RSA) of parallel squares of same size on a planar substrate of area L². In this paper, differently from the classical RSA, the squares obey the size distribution n(a)=n(1)a^−τ, where a=1,2,3,… is the area of the squares. Using this scaling distribution and three classes of packing dynamics we study the final packing fraction of particles, ?(τ,L), and in particular its thermodynamic limit L→∞. We show that the efficiency to attain a high/low packing density of particles on the substrate is strongly dependent on the value of the exponent τ and on the characteristics of the dynamics.     

Crossing Speeds of Random Walks Among “Sparse” or “Spiky” Bernoulli Potentials on Integers

We consider a random walk among i.i.d. obstacles on $\mathbb {Z}$ under the condition that the walk starts from the origin and reaches a remote location y. The obstacles are represented by a killing potential, which takes value M>0 with probability p and value 0 with probability 1?p, 0<p≤1, independently at each site of $\mathbb {Z}$ . We consider the walk under both quenched and annealed measures. It is known that under either measure the crossing time from 0 to y of such walk, τ _y , grows linearly in y. More precisely, the expectation of τ _y /y converges to a limit as y→∞. The reciprocal of this limit is called the asymptotic speed of the conditioned walk. We study the behavior of the asymptotic speed in two regimes: (1) as p→0 for M fixed (“sparse”), and (2) as M→∞ for p fixed (“spiky”). We observe and quantify a dramatic difference between the quenched and annealed settings.     

Three Attractive Osculating Walkers and a Polymer Collapse Transition

Consider n interacting lock-step walkers in one dimension which start at the points {0,2,4,...,2(n?1)} and at each tick of a clock move unit distance to the left or right with the constraint that if two walkers land on the same site their next steps must be in the opposite direction so that crossing is avoided. When two walkers visit and then leave the same site an osculation is said to take place. The space-time paths of these walkers may be taken to represent the configurations of n fully directed polymer chains of length t embedded on a directed square lattice. If a weight λ is associated with each of the i osculations the partition function is $Z_t^{(n)} (\lambda ) = \sum\nolimits_{i = 0}^{\left\lfloor {\tfrac{{(n - 1)t}}{2}} \right\rfloor } {z_{t,i}^{(n)} } \lambda ^i $ where z ⁽ⁿ⁾ _t,i is the number of t-step configurations having i osculations. When λ=0 the partition function is asymptotically equal to the number of vicious walker star configurations for which an explicit formula is known. The asymptotics of such configurations was discussed by Fisher in his Boltzmann medal lecture. Also for n=2 the partition function for arbitrary λ is easily obtained by Fisher's necklace method. For n>2 and λ≠0 the only exact result so far is that of Guttmann and Vöge who obtained the generating function $G^{(n)} (\lambda ,u) \equiv \sum\nolimits_{t = 0}^\infty {Z_t^{(n)} (\lambda )u^t } $ for λ=1 and n=3. The main result of this paper is to extend their result to arbitrary λ. By fitting computer generated data it is conjectured that Z ⁽³⁾ _t (λ) satisfies a third order inhomogeneous difference equation with constant coefficients which is used to obtain $$G^{(n)} (\lambda ,u) = \frac{{(\lambda - 3)(\lambda + 2) - \lambda (12 - 5\lambda + \lambda ^2 )u - 2\lambda ^3 u^2 + 2(\lambda - 4)(\lambda ^2 u^2 - 1){\text{ }}c(2u)}}{{(\lambda - 2 - \lambda ^2 u)(\lambda - 1 - 4\lambda u - 4\lambda ^2 u^2 )}}$$ where $c(u) = \tfrac{{1 - \sqrt {1 - 4u} }}{{2u}}$ , the generating function for Catalan numbers. The nature of the collapse transition which occurs at λ=4 is discussed and extensions to higher values of n are considered. It is argued that the position of the collapse transition is independent of n.