DYNAMICALLY DEFINED SET FOR A DETERMINISTIC RANDOM WALK

The rotor router model is a popular deterministic analogue of a random walk on a graph. Instead of moving to a random neighbor, the neighbors are served in a fixed order. We examine how fast this “deterministic random walk” covers all vertices (or all edges). We present general techniques to derive upper bounds for the vertex and edge cover time and derive matching lower bounds for several important graph classes. Depending on the topology, the deterministic random walk can be asymptotically faster, slower or equally fast compared to the classical random walk.

Jim Propp's rotor–router model is a deterministic analog of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbors in a fixed order.Cooper and Spencer (Comb Probab Comput 15 (2006) 815–822) show a remarkable similarity of both models. If an (almost) arbitrary population of chips is placed on the vertices of a grid ℤd and does a simultaneous walk in the Propp model, then at all times and on each vertex, the number of chips on this vertex deviates from the expected number the random walk would have gotten there by at most a constant. This constant is independent of the starting configuration and the order in which each vertex serves its neighbors.This result raises the question if all graphs do have this property. With quite some effort, we are now able to answer this question negatively. For the graph being an infinite k‐ary tree (k ≥ 3), we show that for any deviation D there is an initial configuration of chips such that after running the Propp model for a certain time there is a vertex with at least D more chips than expected in the random walk model. However, to achieve a deviation of D it is necessary that at least exp(Ω(D2)) vertices contribute by being occupied by a number of chips not divisible by k at a certain time. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010

Jim Propp's rotor–router model is a deterministic analogue of a random walk on a graph. Instead of distributing chips randomly, each vertex serves its neighbours in a fixed order. We analyse the difference between the Propp machine and random walk on the infinite two-dimensional grid. It is known that, apart from a technicality, independent of the starting configuration, at each time the number of chips on each vertex in the Propp model deviates from the expected number of chips in the random walk model by at most a constant. We show that this constant is approximately 7.8 if all vertices serve their neighbours in clockwise or order, and 7.3 otherwise. This result in particular shows that the order in which the neighbours are served makes a difference. Our analysis also reveals a number of further unexpected properties of the two-dimensional Propp machine.

A cylindrical cascade on $\mathbb {T}^d\times \mathbb {R}^r$ can be seen as a deterministic random walk on $\mathbb {R}^r$ driven by an observable over the irrational toral translation on the base torus. We prove that, when the observable is the indicator function of a generic (straight) rectangle in $\mathbb {T}^2$ , the cascade on $\mathbb {T}^2\times \mathbb {R}$ is ergodic for a $G_{\delta }$ -dense set of translation vectors. We also provide examples of ergodic cylindrical cascades in higher dimensions with more restrictive conditions on the side lengths of the rectangles.

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $α$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to $α$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $α$ is quadratic irrational, $β$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of $β$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

Preface. Acknowledgments. Introduction. Probability. Random Walk. Principle of Reflection. Records. Problems that Concern Waiting. Problems that Concern Optimization. Problems on Calculating Probability. Problems on Calculating Expectation. Problems on Statistical Methods. The LAD Method. Probability in Mathematics. Matrix Games. References. Topic Index. Author Index.

Iterative linear equation solvers depend on high quality preconditioners to achieve fast convergence. For sparse symmetric systems arising from large power grid analysis problems, however, preconditioners generated by traditional incomplete Cholesky factorizations are usually of low quality, resulting in slow convergence. On the other hand, although preconditioners generated by random walks are quite effective to reduce the number of iterations, it takes considerable amount of time to compute them in a stochastic manner. In this paper, we propose a new preconditioning technique for power grid analysis, named deterministic random walk, that combines the advantages of the above two approaches. Our proposed algorithm computes the preconditioners in a deterministic manner to reduce computation time, while achieving similar quality as stochastic random walk preconditioning by modifying fill-ins to compensate dropped entries. We have proved that for such compensation scheme, our algorithm will not fail for certain matrix orderings, which otherwise cannot be guaranteed by traditional incomplete factorizations. We demonstrate that by incorporating our proposed preconditioner, a conjugate gradient solver is able to outperform a state-of-the-art algebraic multigrid preconditioned solver on public IBM power grid benchmarks for DC power grid analysis, while potentially remaining very efficient for transient analysis.

By M. Denny and S. Gaines Princeton University Press (2002) pp. 424. ISBN 0-691-09494-2 (pbk) £ 19.95 / $29.95. ![Figure][1] As a teacher of introductory biology, I try very hard to instill in my students the fact that quantitative reasoning is an integral tool

Random walk is a randomly walk process that describes the path including a succession of random steps in the mathematical region. A procedure for estimating the likely position of a point exposed to random motions in probability theory, given the probabilities (which remain constant at each step) of traveling a certain distance in a certain direction. A Markov process that exhibits future behavior independent of prior history is the random walk. This paper will discover the classic recurrence Theorem concerning random walks on the d-dimensional integer lattice, and demonstrate the leads to complete generality with simple methods supported by String’s Approximation. Furthermore, this essay will explore the situation that walker randomly walks in different dimensions and discover that random walk on dimensions one and two will return to the initial point, dimension three and higher dimensions will not. The meaning of this essay is to help more people have some basic understanding with random walk and to know there are some applications of random walk.

We consider in this paper two types of queueing systems which operate under a specific and fixed scheduling policy. The first system consists of a single server and several buffers in which arriving jobs are stored. We assume that arriving parts may require several stages of processing in which case each stage corresponds to a different buffer. The second system is a communication type queueing network given by a graph. The arriving jobs (packets) request a simple path along which they need to be processed. In both models the jobs arrive in a completely deterministic fashion: the interarrival times are fixed and known. All the processing times are also deterministic. A scheduling policy specifies a rule using which arriving parts are processed in the queueing system. A scheduling policy is defined to be stable if there is a finite uniform upper bound on the total number of parts in the system at all times. A necessary condition for stability of any work conserving policy is that the traffic intensity of the station (of each link in the graph in the communication model) is not bigger than one. Many results have demonstrated that this condition is not sufficient for stability. The results were obtained primarily in the context of stochastic networks ([24],[20],[5],[6],[10]), deterministic fluid networks ([6],[9],[7],[8],[2]), deterministic adversarial networks ([4],[1],[14],[12]). One of the earliest result in the area were obtained by Rybko and Stolyar [24] and Lu and Kumar [20]. They showed that a simple priority policy can lead to instability in some queueing networks. Bramson [5] and Seidman [25] showed that even FIFO policy can be unstable in stochastic networks. Instability of FIFO was later demonstrated in an adversarial queueing setting by Andrews et. al. [1]. Dai [6] established that stability of a fluid deterministic queueing network implies stability of a stochastic queueing network. A similar result was established by Gamarnik [12], which connects stability of fluid and adversarial queueing networks. A complete characterization of two-station fluid networks which are stable under any work conserving policy was established by Bertsimas, Gamarnik and Tsitsiklis [2] and Dai and Vande Vate [7]. Goel [14] constructed a complete characterization of adversarial queueing networks which are stable under the usual load condition. The result is extended by Gamarnik [13]. Motivated by a queueing network model stability of homogeneous random walks in a nonnegative orthant was considered in several papers: Malyshev [21], Menshikov [23], Fayolle [11], Ignatyuk and Malyshev [18], Malyshev [22]. Such random walks Q ( t ), t = 0, 1, 2,¡­ have Z + d as a state space ( Z + is the set of nonnegative integers). The transition vectors ¦¤ have deterministically bounded length in max norm and the transition probability p (¦«, ¦¤) along the direction ¦¤ depends only on the face ¦« that the random walk is currently on (the transition probabilities depend only on which coordinates of the current state are positive and which are zero). Such a random walk is defined to be stable if it is positive recurrent. We will also consider deterministic walks, for which p (¦«, ¦¤) is always zero or one (the transition deterministically depends on the face that the walk is currently on). A complete characterization of stable homogeneous random walks in Z + d for d ¡Ü 4 was obtained in Malyshev [21], Menshikov [23] and Ignatyuk and Malyshev [18], but no extension of this classification to higher dimensions has been obtained. Malyshev in [22] establishes a connection between homogeneous random walks and general dynamical systems on compact manifolds and shows that the difficulty of classifying stable random walks is of the same nature as the difficulty of understanding the dynamics of these dynamical systems. Specifically, the complicated dynamics precludes obtaining classification of stable random walks for d = 5. Thus despite many efforts the classification of stable random walks for general dimensions is not known. Likewise classification of stable policies in queueing systems is an open problem.

Random walks with similar transition probabilities

This chapter provides background on different approaches, results, and techniques that are used in several of the chapters. Some of the topics: order statistics and direct integration; random walks and the reflection principle; ballot problems; the Karlin–McGregor theorem; Poisson and compound Poisson approximation; Bonferroni bounds; the Q2/ Q3 Markov-type approximation.

Chapter 8 - Distributed algorithms with dynamical random transitions

A new, simple model of wave propagation for microcellular systems is applied to predict the path loss of a wireless channel. The model does not rely on the classical theory of electromagnetic wave propagation, but it is entirely based on probability theory. We consider the canonical scenario of a random environment of partially absorbing scatterers and model the trajectory of each photon in the system as a random walk. This model leads to a path loss formula that rather accurately (in comparison to other models and experimental data) describes the smooth transition of power attenuation from an inverse square law with the distance to the transmitter to an exponential attenuation as this distance is increased. This result can justify empirical formulas that are often used for path loss prediction, characterized by a breakpoint distance at which the exponent of a power law is increased from a value of approximately 2 to a value in the range of 4 to 10. Theoretical predictions of the model are validated by showing agreement with experimental data collected in the city of Rome, Italy.

Computability of Topological Pressure for Shifts of Finite Type with Applications in Statistical Physics