Ordinary Random Walk Research Articles

Geometric model of quantum navigation during (anti-)search on a plane

A model of joint random walk of two agents in an infinite plane is considered. The agents possess no means of mutual classical communication, but have access to quantum-entanglement resources which are used according to a pre-arranged protocol. The trajectories of the agents diverge on average, but the rate of divergence might be slowed down or speeded up (compared to ordinary random walk) depending on the details of the protocol. This effect has purely quantum nature and can be interpreted in terms of spherical or hyperbolic geometries for slowdown or speedup, respectively.

Lévy movements and a slowly decaying memory allow efficient collective learning in groups of interacting foragers.

Many animal species benefit from spatial learning to adapt their foraging movements to the distribution of resources. Learning involves the collection, storage and retrieval of information, and depends on both the random search strategies employed and the memory capacities of the individual. For animals living in social groups, spatial learning can be further enhanced by information transfer among group members. However, how individual behavior affects the emergence of collective states of learning is still poorly understood. Here, with the help of a spatially explicit agent-based model where individuals transfer information to their peers, we analyze the effects on the use of resources of varying memory capacities in combination with different exploration strategies, such as ordinary random walks and Lévy flights. We find that individual Lévy displacements associated with a slow memory decay lead to a very rapid collective response, a high group cohesion and to an optimal exploitation of the best resource patches in static but complex environments, even when the interaction rate among individuals is low.

Aftermath epidemics: Percolation on the sites visited by generalized random walks.

We study percolation on the sites of a finite lattice visited by a generalized random walk of finite length with periodic boundary conditions. More precisely, consider Levy flights and walks with finite jumps of length >1 [like Knight's move random walks (RWs) in two dimensions and generalized Knight's move RWs in 3D]. In these walks, the visited sites do not form (as in ordinary RWs) a single connected cluster, and thus percolation on them is nontrivial. The model essentially mimics the spreading of an epidemic in a population weakened by the passage of some devastating agent-like diseases in the wake of a passing army or of a hurricane. Using the density of visited sites (or the number of steps in the walk) as a control parameter, we find a true continuous percolation transition in all cases except for the 2D Knight's move RWs and Levy flights with Levy parameter σ≥2. For 3D generalized Knight's move RWs, the model is in the universality class of pacman percolation, and all critical exponents seem to be simple rationals, in particular, β=1. For 2D Levy flights with 0<σ<2, scale invariance is broken even at the critical point, which leads at least to very large corrections in finite-size scaling, and even very large simulations were unable to unambiguously determine the critical exponents.

Random Walk on a Rough Surface: Renormalization Group Analysis of a Simple Model

The field-theoretic renormalization group is applied to a simple model of a random walk on a rough fluctuating surface. We consider the Fokker–Planck equation for a particle in a uniform gravitational field. The surface is modeled by the generalized Edwards–Wilkinson linear stochastic equation for the height field. The full stochastic model is reformulated as a multiplicatively renormalizable field theory, which allows for the application of the standard renormalization theory. The renormalization group equations have several fixed points that correspond to possible scaling regimes in the infrared range (long times and large distances); all the critical dimensions are found exactly. As an example, the spreading law for the particle’s cloud is derived. It has the form R2(t)≃t2/Δω with the exactly known critical dimension of frequency Δω and, in general, differs from the standard expression R2(t)≃t for an ordinary random walk.

Multidimensional random walk with reflections

Reflected random walk in higher dimension arises from an ordinary random walk (sum of i.i.d. random variables): whenever one of the reflecting coordinates becomes negative, its sign is changed, and the process continues from that modified position. One-dimensional reflected random walk is quite well understood from work in 7 decades, but the multidimensional model presents several new difficulties. Here we investigate recurrence questions.

An arcsine law for Markov random walks

The classic arcsine law for the number Nn>≔n−1∑k=1n1{Sk>0} of positive terms, as n→∞, in an ordinary random walk (Sn)n≥0 is extended to the case when this random walk is governed by a positive recurrent Markov chain (Mn)n≥0 on a countable state space S, that is, for a Markov random walk (Mn,Sn)n≥0 with positive recurrent discrete driving chain. More precisely, it is shown that n−1Nn> converges in distribution to a generalized arcsine law with parameter ρ∈[0,1] (the classic arcsine law if ρ=1∕2) iff the Spitzer condition limn→∞1n∑k=1nPi(Sn>0)=ρholds true for some and then all i∈S, where Pi≔P(⋅|M0=i) for i∈S. It is also proved, under an extra assumption on the driving chain if 0<ρ<1, that this condition is equivalent to the stronger variant limn→∞Pi(Sn>0)=ρ.For an ordinary random walk, this was shown by Doney (1995) for 0<ρ<1 and by Bertoin and Doney (1997) for ρ∈{0,1}.

Fluctuation Theory for Markov Random Walks

Two fundamental theorems by Spitzer–Erickson and Kesten–Maller on the fluctuation-type (positive divergence, negative divergence or oscillation) of a real-valued random walk $$(S_{n})_{n\ge 0}$$ with iid increments $$X_{1},X_{2},\ldots $$ and the existence of moments of various related quantities like the first passage into $$(x,\infty )$$ and the last exit time from $$(-\infty ,x]$$ for arbitrary $$x\ge 0$$ are studied in the Markov-modulated situation when the $$X_{n}$$ are governed by a positive recurrent Markov chain $$M=(M_{n})_{n\ge 0}$$ on a countable state space $$\mathcal {S}$$ ; thus, for a Markov random walk $$(M_{n},S_{n})_{n\ge 0}$$ . Our approach is based on the natural strategy to draw on the results in the iid case for the embedded ordinary random walks $$(S_{\tau _{n}(i)})_{n\ge 0}$$ , where $$\tau _{1}(i),\tau _{2}(i),\ldots $$ denote the successive return times of M to state i, and an analysis of the excursions of the walk between these epochs. However, due to these excursions, generalizations of the aforementioned theorems are surprisingly more complicated and require the introduction of various excursion measures so as to characterize the existence of moments of different quantities.

Enhanced diffusion and anomalous transport of magnetic colloids driven above a two-state flashing potential.

We combine experiments and theory to investigate the diffusive and the subdiffusive dynamics of paramagnetic colloids driven above a two-state flashing potential. The magnetic potential was realized by periodically modulating the stray field of a magnetic bubble lattice in a uniaxial ferrite garnet film. At large amplitudes H0 of the driving field, the dynamics of the particle resemble an ordinary random walk with a frequency-dependent diffusion coefficient. However, subdiffusive and oscillatory dynamics at short time scales are observed when decreasing H0. We present a persistent random walk model to elucidate the underlying mechanism of motion, and perform numerical simulations to demonstrate that the anomalous motion originates from the dynamic disorder in the structure of the magnetic lattice, induced by the slightly irregular shape of bubbles.

Weak Ergodicity Breaking of Receptor Motion in Living Cells Stemming from Random Diffusivity

Molecular transport in living systems regulates numerous processes underlying biological function. Although many cellular components exhibit anomalous diffusion, only recently has the subdiffusive motion been associated with nonergodic behavior. These findings have stimulated new questions for their implications in statistical mechanics and cell biology. Is nonergodicity a common strategy shared by living systems? Which physical mechanisms generate it? What are its implications for biological function? Here, we use single particle tracking to demonstrate that the motion of DC-SIGN, a receptor with unique pathogen recognition capabilities, reveals nonergodic subdiffusion on living cell membranes. In contrast to previous studies, this behavior is incompatible with transient immobilization and therefore it can not be interpreted according to continuous time random walk theory. We show that the receptor undergoes changes of diffusivity, consistent with the current view of the cell membrane as a highly dynamic and diverse environment. Simulations based on a model of ordinary random walk in complex media quantitatively reproduce all our observations, pointing toward diffusion heterogeneity as the cause of DC-SIGN behavior. By studying different receptor mutants, we further correlate receptor motion to its molecular structure, thus establishing a strong link between nonergodicity and biological function. These results underscore the role of disorder in cell membranes and its connection with function regulation. Due to its generality, our approach offers a framework to interpret anomalous transport in other complex media where dynamic heterogeneity might play a major role, such as those found, e.g., in soft condensed matter, geology and ecology.

Quasistochastic matrices and Markov renewal theory

Let 𝓈 be a finite or countable set. Given a matrix F = (Fij)i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (qij)i,j∈𝓈, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑n≥0Qn ⊗ F*n associated with Q ⊗ F := (qijFij)i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(Mn, Sn)}n≥0 with discrete recurrent driving chain {Mn}n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Quasistochastic matrices and Markov renewal theory

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Persistence of random walk records

We study records generated by Brownian particles in one dimension. Specifically, we investigate an ordinary random walk and define the record as the maximal position of the walk. We compare the record of an individual random walk with the mean record, obtained as an average over infinitely many realizations. We term the walk ‘superior’ if the record is always above average, and conversely, the walk is said to be ‘inferior’ if the record is always below average. We find that the fraction of superior walks, S, decays algebraically with time, S ∼ t−β, in the limit t → ∞, and that the persistence exponent is nontrivial, β = 0.382 258…. The fraction of inferior walks, I, also decays as a power law, I ∼ t−α, but the persistence exponent is smaller, α = 0.241 608…. Both exponents are roots of transcendental equations involving the parabolic cylinder function. To obtain these theoretical results, we analyze the joint density of superior walks with a given record and position, while for inferior walks it suffices to study the density as a function of position.

Renormalization group for quantum walks

We present a detailed introduction to the discrete-time quantum walk problem, in close analogy with the classical ordinary and persistent random walk. This approach facilitates a uniform application of the renormalization group that highlights similarities and differences between the classical and the quantum walk problem. Specifically, we discuss the renormalization group treatment for the mean-square displacement of a walker starting from a single site on the 1d-line for ordinary and persistent random walks and the quantum walk. We outline the significance of universality for quantum walks and the control this might provide for quantum algorithms. We use our RG method to verify that all 2-state quantum walks on the 1d-line are in the same universality class.

All-time dynamics of continuous-time random walks on complex networks

The concept of continuous-time random walks (CTRW) is a generalization of ordinary random walk models, and it is a powerful tool for investigating a broad spectrum of phenomena in natural, engineering, social, and economic sciences. Recently, several theoretical approaches have been developed that allowed to analyze explicitly dynamics of CTRW at all times, which is critically important for understanding mechanisms of underlying phenomena. However, theoretical analysis has been done mostly for systems with a simple geometry. Here we extend the original method based on generalized master equations to analyze all-time dynamics of CTRW models on complex networks. Specific calculations are performed for models on lattices with branches and for models on coupled parallel-chain lattices. Exact expressions for velocities and dispersions are obtained. Generalized fluctuations theorems for CTRW models on complex networks are discussed.

Maximal-entropy random walk unifies centrality measures

This paper compares a number of centrality measures and several (dis-)similarity matrices with which they can be defined. These matrices, which are used among others in community detection methods, represent quantities connected to enumeration of paths on a graph and to random walks. Relationships between some of these matrices are derived in the paper. These relationships are inherited by the centrality measures. They include measures based on the principal eigenvector of the adjacency matrix, path enumeration, as well as on the stationary state, stochastic matrix, or mean first-passage times of a random walk. As the random walk defining the centrality measure can be arbitrarily chosen, we pay particular attention to the maximal-entropy random walk, which serves as a very distinct alternative to the ordinary (diffusive) random walk used in network analysis. The various importance measures, defined both with the use of ordinary random walk and the maximal-entropy random walk, are compared numerically on a set of benchmark graphs with varying mixing parameter and are grouped with the use of the agglomerative clustering technique. It is shown that centrality measures defined with the two different random walks cluster into two separate groups. In particular, the group of centrality measures defined by the maximal-entropy random walk does not cluster with any other measures on change of graphs' parameters, and members of this group produce mutually closer results than members of the group defined by the ordinary random walk.

Antipersistent dynamics in kinetic models of wealth exchange

We investigate the detailed dynamics of gains and losses made by agents in some kinetic models of wealth exchange. An earlier work suggested that a walk in an abstract gain-loss space can be conceived for the agents. For models in which agents do not save, or save with uniform saving propensity, the walk has diffusive behavior. For the case in which the saving propensity λ is distributed randomly (0≤λ<1), the resultant walk showed a ballistic nature (except at a particular value of λ*≈0.47). Here we consider several other features of the walk with random λ. While some macroscopic properties of this walk are comparable to a biased random walk, at microscopic level, there are gross differences. The difference turns out to be due to an antipersistent tendency toward making a gain (loss) immediately after making a loss (gain). This correlation is in fact present in kinetic models without saving or with uniform saving as well, such that the corresponding walks are not identical to ordinary random walks. In the distributed saving case, antipersistence occurs with a simultaneous overall bias.

Percolation of a collection of finite random walks: a model for gas permeation through thin polymeric membranes

Motivated by recent studies of gas permeation through polymer networks, we consider a collection of ordinary random walks of fixed length l, placed randomly on the bonds of a square lattice. These walks model polymers, each with l segments. Using computer simulations, we find the critical concentration of occupied bonds (i.e., the critical occupation probability) for such a network to percolate the system. Though this threshold decreases monotonically with l, the critical “mass” density, defined as the total number of segments divided by total number of bonds in the system, displays a more complex behavior. In particular, for fixed mass densities, the percolation characteristics of the network can change several times, as shorter polymers are linked to form longer ones.

Quantum walks on a random environment

Quantum walks are considered in a one-dimensional random medium characterized by static or dynamic disorder. Quantum interference for static disorder can lead to Anderson localization which completely hinders the quantum walk and it is contrasted with the decoherence effect of dynamic disorder having strength $W$, where a quantum to classical crossover at time ${t}_{c}\ensuremath{\propto}{W}^{\ensuremath{-}2}$ transforms the quantum walk into an ordinary random walk with diffusive spreading. We demonstrate these localization and decoherence phenomena in quantum carpets of the observed time evolution, we relate our results to previously studied models of decoherence for quantum walks, and examine in detail a dimer lattice which corresponds to a single qubit subject to randomness.

A Biological Interpretation of Transient Anomalous Subdiffusion. II. Reaction Kinetics

Reaction kinetics in a cell or cell membrane is modeled in terms of the first passage time for a random walker at a random initial position to reach an immobile target site in the presence of a hierarchy of nonreactive binding sites. Monte Carlo calculations are carried out for the triangular, square, and cubic lattices. The mean capture time is expressed as the product of three factors: the analytical expression of Montroll for the capture time in a system with a single target and no binding sites; an exact expression for the mean escape time from the set of lattice points; and a correction factor for the number of targets present. The correction factor, obtained from Monte Carlo calculations, is between one and two. Trapping may contribute significantly to noise in reaction rates. The statistical distribution of capture times is obtained from Monte Carlo calculations and shows a crossover from power-law to exponential behavior. The distribution is analyzed using probability generating functions; this analysis resolves the contributions of the different sources of randomness to the distribution of capture times. This analysis predicts the distribution function for a lattice with perfect mixing; deviations reflect imperfect mixing in an ordinary random walk.

A Note on the Transience of Critical Branching Random Walks on the Line

Gantert and Müller (2006) proved that a critical branching random walk (BRW) on the integer lattice is transient by analyzing this problem within the more general framework of branching Markov chains and making use of Lyapunov functions. The main purpose of this note is to show how the same result can be derived quite elegantly and even extended to the nonlattice case within the theory of weighted branching processes. This is done by an analysis of certain associated random weighted location measures which, upon taking expectations, provide a useful connection to the well established theory of ordinary random walks with i.i.d. increments. A brief discussion of the asymptotic behavior of the left- and rightmost particles in a critical BRW as time goes to infinity is provided in the final section by drawing on recent work by Hu and Shi (2008).