Discrete-time Walk Research Articles

Cloud-based mathematical models for self-organizing swarms of UAVs: design and analysis

Unmanned aerial vehicle (UAV) swarms have gained significant attention for their potential applications in various fields. The effective coordination and control of UAV swarms require the development of robust mathematical models that can capture their complex dynamics. The paper introduces mathematical models and relevant paradigms based on the design and analysis of self-organizing swarms of UAVs. The logical and technological construction of the model relies on the theorems developed by authors for obtaining full information exchange during the swarm quasi-random walk. The suggested rotor-router model interprets the discrete-time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on the vertices of the swarm graph. The recommended optimal and fault-tolerant gossip/broadcast schemes support the resilience of swarm to internal failures and external attacks, and cryptographic protocols approve the security. The proposed cloud network topology serves as the implementation framework for the model, encompassing various connectivity options to ensure the expected behavior of the UAV swarms.

The asymptotic shape theorem for the frog model on finitely generated abelian groups

We study the frog model on Cayley graphs of groups with polynomial growth rate D ≥ 3. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

Non-Markovianity and bound states in quantum walks with a phase impurity

We study the discrete-time quantum walk on the line with a single phase impurity. The spread and localisation properties of discrete-time walks initialized at the impurity site are affected by the appearance of bound states and their reflection symmetry. Here, we measure localisation by means of an effective localisation length and an effective participation ratio, which are obtained by averaging over all eigenstates and over all initial states, respectively. We observe that the reduced coin system dynamics undergoes oscillations in the long-time limit with the frequencies determined by the sublattice operator and the bound state quasi-energy differences. The oscillations give rise to non-Markovian evolution, which we quantify using the trace distance and entanglement based measures of non-Markovianity. Indeed, we reveal that the degree of the non-Markovian behaviour is closely related to the emergence of bound states due to the phase impurity. We also show that the considered measures give qualitatively different results depending on the number and symmetries of supported bound states. Finally, comparing localisation and non-Markovianity measures, we demonstrate that the degree of non-Markovianity becomes maximum when the walker is most localised in position space.

One-sided solutions for optimal stopping problems with logconcave reward functions

AbstractIn the literature on optimal stopping, the problem of maximizing the expected discounted reward over all stopping times has been explicitly solved for some special reward functions (including (x+)ν, (ex − K)+, (K − e− x)+, x ∈ ℝ, ν ∈ (0, ∞), and K > 0) under general random walks in discrete time and Lévy processes in continuous time (subject to mild integrability conditions). All such reward functions are continuous, increasing, and logconcave while the corresponding optimal stopping times are of threshold type (i.e. the solutions are one-sided). In this paper we show that all optimal stopping problems with increasing, logconcave, and right-continuous reward functions admit one-sided solutions for general random walks and Lévy processes, thereby generalizing the aforementioned results. We also investigate in detail the principle of smooth fit for Lévy processes when the reward function is increasing and logconcave.

Path-sum solution of the Weyl quantum walk in 3 + 1 dimensions.

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group [Formula: see text], which in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of [Formula: see text] that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e.transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

Hitting statistics from quantum jumps

We define the hitting time for a model of continuous-time open quantum walks in terms of quantum jumps. Our starting point is a master equation in Lindblad form, which can be taken as the quantum analogue of the rate equation for a classical continuous-time Markov chain. The quantum jump method is well known in the quantum optics community and has also been applied to simulate open quantum walks in discrete time. This method however, is well-suited to continuous-time problems. It is shown here that a continuous-time hitting problem is amenable to analysis via quantum jumps: The hitting time can be defined as the time of the first jump. Using this fact, we derive the distribution of hitting times and explicit exressions for its statistical moments. Simple examples are considered to illustrate the final results. We then show that the hitting statistics obtained via quantum jumps is consistent with a previous definition for a measured walk in discrete time [Phys. Rev. A 73, 032341 (2006)] (when generalised to allow for non-unitary evolution and in the limit of small time steps). A caveat of the quantum-jump approach is that it relies on the final state (the state which we want to hit) to share only incoherent edges with other vertices in the graph. We propose a simple remedy to restore the applicability of quantum jumps when this is not the case and show that the hitting-time statistics will again converge to that obtained from the measured discrete walk in appropriate limits.

Quantum search with multiple walk steps per oracle query

We identify a key difference between quantum search by discrete- and continuous-time quantum walks: a discrete-time walk typically performs one walk step per oracle query, whereas a continuous-time walk can effectively perform multiple walk steps per query while only counting query time. As a result, we show that continuous-time quantum walks can outperform their discrete-time counterparts, even though both achieve quadratic speedups over their corresponding classical random walks. To provide greater equity, we allow the discrete-time quantum walk to also take multiple walk steps per oracle query while only counting queries. Then it matches the continuous-time algorithm's runtime, but such that it is a cubic speedup over its corresponding classical random walk. This yields the first example of a greater-than-quadratic speedup for quantum search over its corresponding classical random walk.

A loop reversibility and subdiffusion of the rotor-router walk

The rotor-router model on a graph describes a discrete-time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on vertices of the graph. We prove the following property: if at some moment of time, the rotors form a closed clockwise contour on the planar graph, then the clockwise rotations of rotors generate a walk which enters into the contour at some vertex v, performs a number of steps inside the contour so that the contour formed by rotors becomes anti-clockwise, and then leaves the contour at the same vertex v. This property generalizes the previously proved theorem for the case when the rotor configuration inside the contour is a cycle-rooted spanning tree, and all rotors inside the contour perform a full rotation. We use the proven property for an analysis of the sub-diffusive behavior of the rotor-router walk.

Continuous limit of discrete quantum walks

Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional "coin space" must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on $d$-regular, $d$-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a {\it coined} continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at spatial search by a DTQW over the 2-D torus (a grid with periodic boundary conditions) of size $\sqrt{N}\times\sqrt{N}$, where it was shown \nocite{AAmbainis08} that a coined DTQW can search in time $O(\sqrt{N}\log{N})$, but a standard CTQW \nocite{Childs2004} takes $\Omega(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2-D torus exhibits the $O(\sqrt{N}\log{N})$ scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in \cite{HariKrovi2007}.

The frequency of pattern occurrence in random walks

In the past decade, the use of ordinal patterns in the analysis of time series and dynamical systems has become an important tool. Ordinal patterns (otherwise known as a permutation patterns) are found in time series by taking $n$ data points at evenly-spaced time intervals and mapping them to a length-$n$ permutation determined by relative ordering. The frequency with which certain patterns occur is a useful statistic for such series. However, the behavior of the frequency of pattern occurrence is unstudied for most models. We look at the frequency of pattern occurrence in random walks in discrete time, and we define a natural equivalence relation on permutations under which equivalent patterns appear with equal frequency, regardless of probability distribution. We characterize these equivalence classes applying combinatorial methods. Au cours de la dernière décennie, l’utilisation des motifs ordinaux dans l’analyse des séries chronologiques et systèmes dynamiques est devenu un outil important. Des motifs ordinaux (autrement appelés motifs de permutations) se trouvent dans les séries chronologiques en prenant $n$ points de données au intervalles de temps uniformément espacées et les faisant correspondre à une permutation de longueur $n$ déterminée par leur ordre relatif. La fréquence avec laquelle certains motifs apparaissent est une statistique utile pour ces séries. Toutefois, le comportement de la fréquence d’apparition de ces motifs n’a pas été étudié pour la plupart des modèles. Nous regardons la fréquence d’occurrence des motifs dans les marches aléatoires en temps discret, et nous définissons une relation d’équivalence naturelle sur des permutations dans laquelle les motifs équivalents apparaissent avec la même fréquence, quelle que soit la distribution de probabilité. Nous caractérisons ces classes d’équivalence utilisant des méthodes combinatoires

Logconcave reward functions and optimal stopping rules of threshold form

In the literature, the problem of maximizing the expected discounted reward over all stopping rules has been explicitly solved for a number of reward functions (including (maxfx;0g) , > 0, in particular) when the underlying process is either a random walk in discrete time or a Levy process in continuous time. All of such reward functions are increasing and logconcave while the corresponding optimal stopping rules have the threshold form. In this paper, we explore the close connection between increasing and logconcave reward functions and optimal stopping rules of threshold form. In the discrete case, we show that if a reward function defined on Z is nonnegative, increasing and logconcave, then the optimal stopping rule is of threshold form provided the underlying random walk is skip-free to the right. In the continuous case, it is shown that for a reward function defined on R which is nonnegative, increasing, logconcave and right-continuous, the optimal stopping rule is of threshold form provided the underlying process is a spectrally negative Levy process. Furthermore, we also establish the necessity of logconcavity and monotonicity of a reward function in order for the optimal stopping rule to be of threshold form in the discrete (continuous, resp.) case when the underlying process belongs to the class of Bernoulli random walks (Brownian motions, resp.) with a downward drift. These results together provide a partial characterization of the threshold structure of optimal stopping rules.

Comparing Algorithms for Graph Isomorphism Using Discrete- and Continuous-Time Quantum Random Walks

Berry and Wang [Phys. Rev. A {\bf 83}, 042317 (2011)] show numerically that a discrete-time quantum random walk of two noninteracting particles is able to distinguish some non-isomorphic strongly regular graphs from the same family. Here we analytically demonstrate how it is possible for these walks to distinguish such graphs, while continuous-time quantum walks of two noninteracting particles cannot. We show analytically and numerically that even single-particle discrete-time quantum random walks can distinguish some strongly regular graphs, though not as many as two-particle noninteracting discrete-time walks. Additionally, we demonstrate how, given the same quantum random walk, subtle differences in the graph certificate construction algorithm can nontrivially impact the walk's distinguishing power. We also show that no continuous-time walk of a fixed number of particles can distinguish all strongly regular graphs when used in conjunction with any of the graph certificates we consider. We extend this constraint to discrete-time walks of fixed numbers of noninteracting particles for one kind of graph certificate; it remains an open question as to whether or not this constraint applies to the other graph certificates we consider.

Finite size Lyapunov exponent for some simple models of turbulence

Exact expressions for the finite size Lyapunov exponent λ(δ) are found and analyzed for several idealized models of turbulence in 1D and 2D. Among them are a random walk with discrete time and continuously distributed jumps and an isotropic Brownian flow in 2D also known as the Kraichnan flow. For the former a surprising fact is a δ−1 scaling for intermediate values of δ in contrast to δ−2 well known for a random walk in continuous time (Brownian flow) and for a simple random walk in discrete time. For the Kraichnan flow an exact relation is established between the scaling of λ(δ) and the scaling of relative dispersion in time.

Exact encounter times for many random walkers on regular and complex networks

The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for independent walkers, where any number of walkers can occupy the same site, and for walkers with an exclusion interaction, when no site can contain more than one walker. These analytical results are then compared with numerical simulations, showing very good agreement.

Statistics of the occupation time for a random walk in the presence of a moving boundary

We investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v. For the continuous-time problem (Brownian motion), we provide a simple alternative proof of Newman's recent result [J.Phys.A 34, L89 (2001)] using a method due to Kac. We then discuss the same problem for the case of a random walk in discrete time with an arbitrary distribution of steps, taking advantage of the general set of results due to Sparre Andersen. For the binomial random walk we analyse the corrections to the continuum limit on the example of the mean occupation time. The case of Cauchy-distributed steps is also studied.

Random vicious walks and random matrices

A lock step walk is a one-dimensional integer lattice walk in discrete time. Suppose that initially there are infinitely many walkers on the nonnegative even integer sites. At each moment of time, every walker moves either to its left or to its right with equal probability. The only constraint is that no two walkers can occupy the same site at the same time. Hence we describe the walk as vicious. It is proved that as time tends to infinity, a certain limiting conditional distribution of the displacement of the leftmost walker is identical to the limiting distribution of the (scaled) largest eigenvalue of a random GOE matrix (GOE Tracy-Widom distribution). The proof is based on the bijection between path configurations and semistandard Young tableaux established recently by Guttmann, Owczarek, and Viennot. The distribution of semistandard Young tableaux is analyzed using the Hankel determinant expression for the probability obtained from the work of Rains and the author. The asymptotics of the Hankel determinant are then obtained by applying the Deift-Zhou steepest-descent method to the Riemann-Hilbert problem for the related orthogonal polynomials. © 2000 John Wiley & Sons, Inc.

The continuous-time random walk description of photon motion in an isotropic medium

Lattice random walk models are useful alternatives to diffusion theory for describing photon migration in turbid media. Results obtained from random walk models were, in the past, based on approximating the propagator as a Gaussian function. This is equivalent to assuming that a large number of steps have been taken. We show that a particular, physically plausible, continuous-time random walk (CTRW) model can be solved exactly. At sufficiently long times the solution reduces to that for a random walk in discrete time. We provide exact solutions to problems related to cw reflection measurements and indicate some generalizations to time-gated measurements made on a uniform medium.

Asymptotic distributions for the Ehrenfest urn and related random walks

The distributions of nearest neighbour random walks on hypercubes in continuous time t 0 can be expressed in terms of binomial distributions; their limit behaviour for t, N → ∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity for N →∞. Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.

Asymptotic distributions for the Ehrenfest urn and related random walks

The distributions of nearest neighbour random walks on hypercubesin continuous timet0 can be expressed in terms of binomial distributions; their limit behaviour fort, N →∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity forN →∞.Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.

Size-dependent diffusion coefficient in a myopic random walk on a strip

We consider a random walk in discrete time ( n = 0, 1, 2, …) on a square lattice of finite width in the y-direction, i.e., {j, m | j ϵ Z, m = 1, 2, 3, …, N} . A myopic walker at ( j,1) or ( j, N) jumps with probability 1 3 to any of the available nearest-neighbor sites at the end of a time step. This couples the motions in the x- and y-directions, and leads to several interesting features, including a coefficient of diffusion in the x-direction that depends on the transverse size N of the strip. Explicit solutions for 〈 x 2 n 〉 (and the lateral variance 〈 y 2 n 〉) are given for small values of N. A closed-form expression is obtained for the (discrete Laplace) transform of 〈 x 2 n 〉 for general N. The asymptotic behaviors of 〈 x 2 n 〉 and 〈 y 2 n 〉 are found, the corrections falling off exponentially with increasing n. The results obtained are generalized to a myopic random walk in d dimensions, and it is shown that the diffusion coefficient has an explicit geometry dependence involving the surface-to-volume ratio. This coefficient can therefore serve as a probe of the geometry of the structure on which diffusion takes place.